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UBC Theses and Dissertations

Local radical and semi-simple classes of rings Stewart, Patrick Noble 1969

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L O C A L R A D I C A L AND S E M I - S I M P L E  CLASSES  OP R I N G S  by  PATRICK B.A., M.A.,  University  University  NOBLE of  of  B r i t i s h Columbia,  California at  A T H E S I S SUBMITTED  IN  THE R E Q U I R E M E N T S  the  I965  Berkeley,  PARTIAL FULFILMENT FOR THE  DOCTOR OP in  STEWART  DEGREE  1966  OF  OF  PHILOSOPHY  Department of  MATHEMATICS We a c c e p t required,  this  thesis  conforming to  standard'  T f f i T t J N IV E R S I T Y  OF'BKlTlSH  July, 1  as  1969  COLUMBIA  the  In  presenting  this  an a d v a n c e d d e g r e e the I  Library  further  for  agree  in  at  University  the  make  that  it  partial  freely  this  representatives. thes,is  for  It  financial  gain  of  The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, Canada  of  Columbia,  British for  by  the  Columbia  shall  not  the  requirements  reference copying of  Head o f  understood that  written permission.  Department  of  extensive  granted  is  fulfilment  available  permission for  s c h o l a r l y p u r p o s e s may be  by h i s of  shall  thesis  I agree and  for  that  Study.  this  thesis  my D e p a r t m e n t  or  copying or'.publication  be a l l o w e d w i t h o u t  my  Supervisor:  N.  J.  Divinsky ii  ABSTRACT  For any c l a s s of r i n g s class  ft(K)  subring  K >_ 2  c a r d i n a l number  ft  we  R  any  non-empty  make the f o l l o w i n g d e f i n i t i o n s .  i s the c l a s s of a l l r i n g s  of  and  R  The  •  such that every  which i s generated by a s e t of c a r d i n a l i t y  s t r i c t l y l e s s than  K  i s i n ft .  c l a s s of a l l r i n g s  R  such t h a t every non-zero homomorphic  image of  R  contains  The  class  a non-zero s u b r i n g  R  in  g(K)  ft  ^  x S  which i s  generated by a s e t of c a r d i n a l i t y s t r i c t l y l e s s than Several properties are determined.  IT  ft(K)  Necessary and  of a l l  R  classes  In p a r t i c u l a r , c o n d i t i o n s  which imply that class.  of the  g(K)  R g  e  (K)  K . a  n  d  are s p e c i f i e d  i s a r a d i c a l c l a s s or a semi-simple s u f f i c i e n t conditions  semi-simple r i n g s be  t h a t the  equal to  3"(K)  class are  given. The K = £|  are  classes  ft(K)  considered  v  and  1  cases when  cal  In a l l cases when  ft  ft  K = 2 classes  ft(2)  i s one  ft(2)  and  special r a d i c a l classes. which are  only i f f o r a l l  contained  x e R , x  in  or ft  of the well-known r a d i of the well-known ft(}»^ ) o  whenever they c o n t a i n a l l n i l p o t e n t r i n g s  are shown to be classes  / l M  i s one  r a d i c a l c l a s s e s i t Is shown that c l a s s e s and  when  g(K)  i n d e t a i l f o r various  ( i n c l u d i n g the classes).  ft  PC  are r a d i c a l they  Those r a d i c a l (R € FC  i s t o r s i o n ) are  i f and  characterized.  iii Let  M(H )  class (0) of  ft  be any r a d i c a l c l a s s .  ( i f one e x i s t s )  0  f o r a l l rings ft  R  ft  radical  W(H )( ) ( ) R  such t h a t  n a R  =  0  i s d e f i n e d t o be the l o c a l complement  and i s denoted by ft .  complement  The l a r g e s t  e x i s t s and ft =  I f ft = (K ) R  0  ft(2)  .  then the l o c a l  The l o c a l complements  of a l l r a d i c a l s d i s c u s s e d a r e determined. We a r e a b l e t o a p p l y some of these r e s u l t s i n order to c l a s s i f y those c l a s s e s and r a d i c a l  classes.  of r i n g s which are both semi-simple  iv  TABLE OP CONTENTS Page INTRODUCTION CHAPTER  I 1.1 1.2  CHAPTER  1  PRELIMINARIES R a d i c a l Theory. Rings Without N i l p o t e n t Elements  • • -  4 9  I I K-CLASSES AND GENERALIZED K-CLASSES 2.1 2.2 2- 3 2.4  K-Classes L o c a l Classes Elementary Classes Generalized K-Classes  12 16 32 38  CHAPTER I I I ELEMENTARY RADICAL CLASSES 3.1 3.2 3- 3 3.4 CHAPTER  The Elementary R a d i c a l Classes tr , tr and PC. 57 The Elementary R a d i c a l Classes $' and £>' . 62 Classes f o r which H' = "fl 65 Elementary R a d i c a l Classes which are <_ PC . . 67 R  IV GENERALIZED ELEMENTARY AND LOCAL RADICAL CLASSES 4.1 4.2 4.3 4.4  Absorbent C a r d i n a l Numbers Generalized R a d i c a l Classes which are >_ y\ Generalized R a d i c a l Classes which are <_ PC . The Generalized R a d i c a l Class e l LOCAL RADICAL CLASSES  85 92 98 107  The L o c a l R a d i c a l Classes and £ . The L o c a l R a d i c a l Classes J*, and P I * . L o c a l R a d i c a l Classes M f o r which £ <_ 3* <_7| L o c a l Complementary R a d i c a l Classes . A Representation of &' as the I n t e r s e c t i o n of R a d i c a l Classes . Semi-Simple R a d i c a l Classes  114 118 129 135  BIBLIOGRAPHY  I58  s  CHAPTER  V 5.1 5.2 5.3 5.4 5.5 5.6  147 151  V o  ILLUSTRATIONS Page 1.  Elementary R a d i c a l Classes  2.  G e n e r a l i z e d R a d i c a l Classes  3.  Local Radical Classes  4.  The Classes  5.  I n t e r s e c t i o n s o f L o c a l R a d i c a l Classes  6.  I n t e r s e c t i o n s of L o c a l R a d i c a l Classes and  7.  Summary of L o c a l R a d i c a l Classes  J  ,  $*  84 . .  . . . . .  125  . . . . . , j'  and  113  .  J  128 14-7  ' 150  CRH(y) .) e  .  157  vi  A CKNOWLEDGEMENTS  The Dr.  N. J . D i v i n s k y ,  preparation  of this  The typing for  author wishes  f o r advice thesis.  draft  typing the f i n a l  supervisor,  and encouragement d u r i n g t h e  author a l s o wishes  the f i r s t  The  t o thank h i s r e s e a r c h  t o thank h i s wife f o r  of the t h e s i s ,  and Mrs. Monlisa  Wang  draft.  financial  support  C o u n c i l o f Canada i s g r a t e f u l l y  o f the N a t i o n a l acknowledged.  Research  - 1 -  INTRODUCTION  The cal  classes  conditions will if  purpose of t h i s  and s e m i - s i m p l e on f i n i t e l y  special  generated  generated  condition.  elementary class ring  classes  be c a l l e d a l o c a l c l a s s  every f i n i t e l y  of  R  thesis  i s to Investigate  which a r e determined by  subrings.  i fa ring subring of  Reft  A class  Reft R  S i m i l a r i l y , a class  i fa ring  radi-  o f r i n g s ft  i f and o n l y  s a t i s f i e s some  of rings  ft  i s an  i f and o n l y i f e v e r y s u b -  g e n e r a t e d b y one e l e m e n t s a t i s f i e s some  special  condition. Let  u s c o n s i d e r some e x a m p l e s .  of  a l l commutative r i n g s  is  commutative  ring  of  R  i f and o n l y i f e v e r y f i n i t e l y  class  since a r i n g  We  y  This  rings  R  x  that  of rings  I s n o t an  x e R  class the  of a ring  element.  involves  elements.  This  a l l Jacobson r a -  Notice  (for a l l  x + y + xy = 0)  of a n i l ring  of r i n g  i s an elementary  the c l a s s - o f  i s not a l o c a l c l a s s .  quantification  cation  R  is nil.  s h a l l prove  such that  finition  a ring  generated sub-  i s n i l i f and o n l y i f f o r a l l  of a Jacobson r a d i c a l r i n g  e R  class  of a l l n i l rings  s u b r i n g g e n e r a t e d by  tion  since  the class  class. The  dical  i s a l o c a l class  i s commutative.  elementary  Clearly  that  the d e f i n i -  x e R  there i s a  involves  the e x i s t e n t i a l  On t h e o t h e r hand t h e d e -  only  the u n i v e r s a l  i l l u s t r a t e s what we  quantifiwould  - 2 -  n a t u r a l l y expect : by  conditions  t h a t c l a s s e s of r i n g s which are d e f i n e d  i n v o l v i n g only u n i v e r s a l q u a n t i f i c a t i o n of  r i n g elements would be l o c a l c l a s s e s whereas c l a s s e s o f r i n g s which are d e f i n e d by c o n d i t i o n s  involving e x i s t e n t i a l quanti-  f i c a t i o n o f r i n g elements would not be l o c a l c l a s s e s . example, we would expect that the c l a s s o f a l l r i n g s f y i n g a given  For satis-  s e t of p o l y n o m i a l i d e n t i t i e s would be a l o c a l  class. In Chapter I I we c o n s i d e r about l o c a l c l a s s e s .  some g e n e r a l  results  In p a r t i c u l a r , we s p e c i f y s e v e r a l  con-  d i t i o n s under which l o c a l c l a s s e s are r a d i c a l c l a s s e s o r semisimple  classes. The  remainder of the t h e s i s i s devoted to a c o n s i -  d e r a t i o n o f s p e c i f i c l o c a l r a d i c a l c l a s s e s and s p e c i f i c semi-simple  local  classes.  Any  class of rings  ft  determines a l o c a l and an  elementary c l a s s (the c l a s s o f a l l r i n g s such t h a t every f i n i t e l y generated s u b r i n g i s i n ft). We c o n s i d e r  ( s u b r i n g generated by one element)  the l o c a l and elementary c l a s s e s de-  termined by the well-known r a d i c a l c l a s s e s .  A l l o f these  c l a s s e s a r e r a d i c a l c l a s s e s and those which c o n t a i n a l l n i l potent r i n g s are shown to be s p e c i a l r a d i c a l c l a s s e s .  In  t h i s and i n other ways we a r r i v e a t s e v e r a l new r a d i c a l classes. are  A l l those which a r e elementary c l a s s e s and which  contained  that f o r a l l  i n FC (FC x e R , x  i s the c l a s s o f a l l r i n g s  R  such  i s t o r s i o n ) can be c h a r a c t e r i z e d as  - 3 -  "sums" o f c e r t a i n s i m p l e e l e m e n t a r y r a d i c a l c l a s s e s .  I n some  cases we a r e a b l e t o o b t a i n s t r u c t u r e theorems by assuming c e r t a i n chain Any  conditions. c l a s s o f r i n g s which i s c l o s e d under homomor-  p h i c images ( a c t u a l l y a s l i g h t l y weaker c o n d i t i o n i s s u f f i c i e n t here) determines an e l e m e n t a r y and a l o c a l s e m i - s i m p l e class.  These c l a s s e s a r e i n v e s t i g a t e d i n Chapter I V . F o l l o w i n g A n d r u n a k i e v i c [ 2 ] we d e f i n e l o c a l com-  p l e m e n t a r y r a d i c a l s and determine t h e l o c a l complements o f the r a d i c a l s which a r e d i s c u s s e d . F i n a l l y we a r e a b l e t o a p p l y some o f o u r r e s u l t s i n - o r d e r t o c l a s s i f y a l l c l a s s e s which a r e b o t h s e m i - s i m p l e and  r a d i c a l classes.  - k -  CHAPTER I PRELIMINARIES 1 . 1 RADICAL THEORY: In this thesis we shall use the following notational conveniences; (1)  Let R be a ring: (i) If S c R , <S> = the subring of R generated by the elements of S . (ii)  If x ,  x c R , <x  ±  1}  N  x> = N  < C 2' ' • • •» ^ * If S c R , (S) = ideal of R generated by the x  (iii)  R  elements of S . (iv) If  x ,  x  1  N  e R ; (x.^  x ) = N  R  I <J R for "I is an ideal of R . "  (2)  We shall write  (3)  Classes of rings will usually be denoted by script letters and a l l classes of rings are assumed to be non-empty.  (4)  If ft and H are two classes of rings we shall W <_ ft for " M is contained in ft".  write (5)  Two classes ft and ft <  11  nor  are unrelated i f neither  W < ft .  Let ft be a class of rings. We list several  - 5 -  c o n d i t i o n s which (A)  If  Re  then (B)  (D)  ft  ft  may  and  satisfy:  R'  i s a homomorphic  F o r any r i n g  R  there  e x i s t s ft(R) <3 R  ft(R) e ft a n d i f  J < R  I f every non-zero  homomorphic  Every non-zero phically  (F)  and  ideal  ideal  ideal  ring  And i f ft s a t i s f i e s  1 . 1 . 1 (i)  F o r any r i n g  If  ft  of  (ill)  R  ,  in  R  c a n be homomor-  ring  i n ft .  c a n be  in  ft  (0)  =  i s any c l a s s R  such that  of rings,  homomorphically  then  R e f t . satisfy:  .  l e f t ,  and  i s an i d e a l o f I  is a  ft-ideal  R .  A class  of rings  if  ft  satisfies  If  ft  is  rings the a  I  then  ft  Is  a radical  conditions  a radical  class  G  class  i s a semi-simple of a l l  U s.s.  a class  of rings  class  i f and o n l y  ( A ) , (B) a n d (C) .  then  (ft s.s.) i f a n d o n l y i f ft(R) =  (iv)  &  R  R e f t .  ( B ) , i t may a l s o  ft(R/ft(R))  J c ft(R) .  DEFINITION:  a ring  (ii)  i n ft , t h e n  of  that  image o f a r i n g  mapped o n t o a n o n - z e r o  I f every non-zero  such  J e ft t h e n  of a r i n g  mapped o n t o a n o n - z e r o  (C)  R  R' e ft .  contains a non-zero (E)  image o f  class rings  R  i s ft s e m i - s i m p l e  (0)  .  A class  of  i f and o n l y i f f o r some r a d i c a l  G = class  .  I f Hi  is  satisfying  (£),  = the  - 6 -  class  of a l l rings  R  which  mapped o n t o a n o n - z e r o (v)  A class  of rings  ft  ring  c a n n o t be  homomorphically  in_77i •  i s hereditary  i f whenever  I < R e f t , I e f t . (vi)  and (a)  If  R e  (b)  Wi  is  (c)  If  R e  The c a n be f o u n d  then  of rings i f  R  i s a prime  ring.  hereditary. and  R  i s an i d e a l o f a r i n g  K / ( 0 : R ) € WI  = (0))  Rx  where  (0:R) = {x e K  i n Rings  K , : xR =  .  above d e f i n i t i o n s a n d t h e f o l l o w i n g and R a d i c a l s  theorems  b y N. J . D i v i n s k y  [7].  THEOREM: A class  of rings  o n l y i f ft s a t i s f i e s  1.1.3  ft  i s a radical class  i f and  (A) a n d ( D ) .  THEOREM: If  IMyft  ytfl  i s a class  i s a radical class.  will  refer  m  .  1.1.4  class  only i f :  then  1.1.2  i s a special  o f r i n g s ffl  A class  to  of rings When  as the upper  which  '\JLy\ft,  satisfies  (E) t h e n  i s a radical class  radical class  determined by  THEOREM: If  W  i s a r a d i c a l class  we  then the c l a s s  of a l l  - 7 H s.s.  rings  i f Yfl  is a  t h e n yY[  satisfies class  — the  semi-simple (E)  and  1.1.5  of  rings  class  class  conditions  of  and  Conversely,  s.s.  only i f  rings.  G  (E)  Thus,  satisfies  and  (P)  G  is  a  conditions  (P).  DEFINITION: M  is a  special radical class  1/ty^for some s p e c i a l c l a s s o f r i n g s  1.1.6  i f and  only i f  tt  =  •  THEOREM: If  M  potent rings if  (F).  s a t i s f y i n g conditions  a l l  i f and  (E)  and  i s a hereditary  are tt  only i f  tt  in  tt  than  =\Jl,y^ where  radical class  and  no n i l -  is a special radical =  the  class  of  class  prime  tt s .s . r i n g s .  1.1.7  THE  LOWER RADICAL CONSTRUCTION. If  tt^ =  tt  the of  And  Hp  class rings  f o r any  limit -  i s any of in  class  of  a l l rings tt  rings,  which are  define: homomorphic images  .  o r d i n a l number  /3 >_ 2  , if  0  i s not  a  ordinal: the  class  of  a l l rings  R  z e r o homomorphic image o f ideal If  we  j3  is a  , £-1 ordinal:  which i s i n limit  S  fl  .  such that R  every  contains a  non-  non-zero  - 8 -  •Hp  = U(W  Let is  ¥  =  (i)  UCMg  : j3  is  a n o r d i n a l number}  We w i l l  determined  by  refer the  to.  .  H  class  as  Then the  W  lower  M .  DEFINITION: Let  N  R  be  of  a ring,  integer; if  and  that  I  = I  A subring  I  of  such  R  R  if  and  N  of  R  In  only for  order  radical  discussed  in Divinsky  FF = the  upper  finite = the  I  a subring  I  1  is  R  and  N  a  subring  of  degree  only, i f  there  exists  1^,...,  1^  c  <  ...<  I  R  an a c c e s s i b l e  I  2  <  a ring I  is  is  establish  classes  a l l  N  A R  integer  the  but  the  first  subring  subring  N  notation  R  .  an a c c e s s i b l e  some p o s i t i v e  radical  of  an a c c e s s i b l e  i f  to  following  F  •  class.  class  positive  (ii)  Y < 13}  a radical  radical  1.1.8  :  y  of  degree  .  we  list  the  of  which  are  [7]• class  determined  by  the  class  of  a l l  radical  class  determined  by  the  class  of  a l l  radical  class  determined  by  the  class  of  full  determined  by  the  class  of  a l l  determined  by  the  class  of  a l l  fields.  upper  fields. 3" = t h e  upper  matrix the  upper  simple ti=  the  rings  radical  rings  upper  over  with  radical  division class  rings.  unity. class  of  - 9 -  simple n o n - t r i v i a l J = the upper primitive j8  = the upper  radical class  rings  £ = the c l a s s  radical class Irreducible  nilpotent  of a l l  w i t h idempotent  hearts.  determined by t h e c l a s s  of a l l  of zero.  of a l l n i l rings. of a l l l o c a l l y  nilpotent  rings.  determined by the c l a s s  of a l l  determined by the c l a s s  of a l l  rings.  the lower r a d i c a l c l a s s nilpotent  rings  N  such that  w i t h D.C.C. o n l e f t  cj* = t h e l o w e r r a d i c a l c l a s s zero simple  1.2  determined by the c l a s s rings  radical class  j3 = t h e l o w e r r a d i c a l c l a s s  R  of a l l  without proper d i v i s o r s  = the c l a s s  ^ =  determined by the c l a s s  rings.  subdirectly = the upper  rings.  N = "/^(R)  f o r some  ring  ideals, d e t e r m i n e d by t h e c l a s s  of a l l  rings.  RINGS WITHOUT NILPOTENT ELEMENTS. Our p u r p o s e  1.2.1  i s to establish:  THEOREM: A ring  to  i n this section  a subdirect  R  without nilpotent  sum o f r i n g s  elements  i s isomorphic  without proper d i v i s o r s of zero.  • I t w i l l be c o n v e n i e n t t o f i r s t  prove:  x  - 10 -  1.2.2  LEMMA: If  then  R  (i) x x  h a s no n i l p o t e n t  = [ y e R : x y = 0} <1 R  r  = x  r  x $ x  (ii) (iii) (iv)  elements  0 ^ x e R  and  and  = ( y e R : y x = 0} ,  t  ,  t  i f r e R  and  the f a c t o r  ring  r x € x^ R/ ^ x  n  then a  s  n  0  r e x^ ,  nilpotent  elements  Proof: Let  R  be a r i n g  w i t h no n i l p o t e n t  elements and 2  O ^ x e R . xa = 0 . blishes and (ab)  a e R  = 0  =0  and  Similarily i f  ( i ) . Since  ab 2  If  so  then  x  2  ax = 0  xa = 0  then  then  (xa) = 0  ax = 0 :  ^ 0 , ( i i )i s clear.  (bab) = 0  ab = 0 .  so  Prom t h i s  .  so  this  esta-  I f a,b e R  bab = 0 , b u t t h e n ( i i i ) and ( i v ) f o l l o w  immediately. Q.E.D. To p r o v e each non-zero R/I(x) Z(x) has  the theorem i t i s s u f f i c i e n t t o f i n d , f o r  x e R , an i d e a l  h a s no p r o p e r d i v i s o r s  I(x) of  o f zero and  = ( I < R : x. $ I , i f r x e I no n i l p o t e n t  Z(x)  $  chain i n Zorn's  elements}  and i t i s c l e a r Z(x) i s also  Lemma, I ( x ) If  a e R  .  B y 1.2.2  that  in  f o r which  x £ I(x)  . Let  r e I , and R/I x^ e Z(x)  so  the union o f an ascending  Z(x) .  maximal i n and  then  R  Thus we may c h o o s e , b y  Z(x).  a £ l(x) l e t  - 11  J  = (y  in  e R  R/I(x)  R/I(x) rx  : ay  .  e J  and by Since  then  Finally  e l(x))  by  l(x) e l(x)  l,2.2.(iv)  elements,  so  no p r o p e r  divisors  x  ( )  J e Z(x) of  •  x  (a  1.2.2(1)  a  arx  2  -  ,  ax  so  T  h  e  J/I(x)  n  + I(x)), <£ l ( x )  ar  =  so  e l(x)  ,  R/J = R/I(x)/J/I(x) .  Hence  J = l(x)  = (a +  I(x))  (a + I ( x ) ) x  <t J  hence  in  .  If  r  e J  has  no  so  R/l(x)  p  .  nilpotent has  zero.  Q.E.D.  This and R j a b u h i n In [3]  result  R  is  terms  of  the  been  [14] we  radical  elements.  proven  i f  by  involving c l a s s "/}  can r e s t a t e  semi-simple  ~Ylcr o  potent  also  [4] u s i n g a n a r g u m e n t  and T h i e r r i n  ring  has  1.2.1  and o n l y  Andrunakievic m-systems. of  as i f  Andrunakievic  follows: R  has  •A  no n i l -  - 12  -  CHAPTER  II  K - C L A S S E S AND G E N E R A L I Z E D K - C L A S S E S  .1  K-CLASSES: We b e g i n  2.1.1  only  2.1.2  i f  a l l  (i)  any  and  only  and  the  ft  of  is  strongly  rings  in  c a r d i n a l number  K  (iii)  if  class  a  R  is  such of  class  Some  is  R  a  ft  hereditary,  are  in  ft  i f  .  a  rings  that  rings of  A  K-ring i f  of  rings  a K-subring of  A c  S  is  £ K  R ,  every  7  immediate  ft  ,  is  set  c a r d i n a l i t y of  For any  is  a ring  there  R  A class  the  S of  A ring  rings  is ft(K)  such  a  a K-class  ft  such  that  consequences  of  that  i f  <A>  = S  . K-subring of  is  the  K-subring of  is  R  i f  class  of  . all.  is  in  ft  and o n l y  i f  there  3 = these  R  R  ft(K)  .  .  definitions  following:  PROPOSITION: Let  be  rings  subrings  A subring  (ii)  2.1.3  of  DEFINITION: For  are  following_definitions.  DEFINITION: A class  and  w i t h the  cardinal  U  and  numbers.  ft  be  classes  of  rings  and  K  and  T  - 1 3 -  (i)  34(K)  i s . s t r o n g l y h e r e d i t a r y and  Jii)  If  34 <. R  then  M(K) <. R(K) .  (iii)  If  K <. T  then  H(r)  (iv) (v)  34 i s a K - c l a s s  < »(K) = 34(K)(T) .  i f and only i f  34 = 34(K) .  I f s u o d i r e c t sums o f r i n g s i n s u b d i r e c t sums o f r i n g s i n  ' (vi) I f  «(K) < 34(K)(T) .  34 s a t i s f i e s c o n d i t i o n  34 are i n  34  34(K) a r e i n (A), so does  then  34(K) . 34(K) .  Proof: (i)  If  S  i s a s u b r i n g of a r i n g  are subrings and  R e  of  34(K) ,  T-subrings of (ii)  Assume t h a t subring of R e  (iii)  R  i s a subring of  »(K)  R e  so  S  R  M(K)(r) .  Re  »(K) .  S e 34 <_ R so  K < r . R  then  K <_ T .  If S  Thus  Re  If  S  i s a K-  S e R  .  As above,  S  S e 34 s i n c e  S e 34 so  is a S  and  S  i s a K-ring.  and hence by ( i )  34. i s a K - c l a s s .  of r i n g s  such t h a t  34(K) = ft(K)(K) .  and  Re  T-subring  Assume t h a t R  34(r)  S  Hence,  i s a K-  i s a l s o a T-subring  R e 34(K)(T)  M(K)(r) <_ 34(K) (iv)  of  W(K) ; and i n p a r t i c u l a r , a l l  R. a r e i n  then  Let  Then  S €  then subrings  R(K) .  subring of  R .  So i f S  34 <. R and  Assume t h a t  since  R .  R  of  R  34(K) . be a K-subring of  of  R  so  S e 34(K).  Therefore,  34(K)(r) = 34(K) .  Then there i s a c l a s s  34 = ft(K) .  Therefore,  However, by ( i i i ) R(K)(K) = R(K) =34,  -14  -  s o tt = .Jt(K) . (v)  Suppose that  R  i s a r i n g with ideals  n(I  : a e A } = (0)  and  I  R/I  e  be a K - s u b r i n g o f  R  : a e A  such  tt(K)  for a l l  e A .  a  Let S  S  +  I  S  .  Then  a  = -j—  i s a K-subring  e » •  of  fl(S n I  Moreover,  R  / I  a  , hence  : a e A } = (0)  Q  s o by  a on M  our assumption (vi)  Assume t h a t image o f  tt(K)  R e  R  .  , S e si  Let  S'  R'  is a  be a K - s u b r i n g S  homomorphic  image o f  S  the  {x^}  or .  R  such  ( i f S'  determined  by  subring generated  by  (x } .)  S e t t  and s i n c e  »  satisfies  fore,  R'  R e tt(K) .  Therefore,  and t h e  there Is a K-subring  cosets  .  homomorphic  of  R  that  S  .  7  Then  is a  7  i s generated  (x ) a  Since  let R e  (A), S'  S  be t h e  tt(K)  e M  by  .  , There-  e »(K) .  Q.E.D. I n P r o p o s i t i o n 2.1-3 conditions  on  tt  are i n h e r i t e d  it  i s not true that  tt  is  a radical  situation  ( v ) - ( v i ) we  tt(K)  class.  tt(K)  by  .  must be a r a d i c a l  The n e x t  c a n n o t o c c u r when  K >  see t h a t  Unfortunately, class  t h e o r e m shows t h a t ^  Q  .  some  whenever this  -  2.1.4  -  15  THEOREM: If  radical  K  i s a c a r d i n a l number  c l a s s then  tt(K)  Jf  £  i s a radical  a  Q  n  d  w  l  s  a  class.  Proof: Assume t h a t First is  £ K  set  .  A = T £ K  S  .  of  A  <A>  A  and  .  The c a r d i n a l i t y  A—±_B  class. then  A/B  e  I  + J e M(K)  are  o  R  there  S <  B <1 A  .  Let  A'  and t h a t  be a K - s u b r i n g  A' n B < t h e c a r d i n a l i t y o f B  .  Since  - — g — ^ i s a K-subring of A  e  is a  r £ K' .  i s a ring,  M(K)  of  • /  w  of K-subrings  because  M  that  i f  A  and  U  is a  A/B  B e W(K), A/B e M ( k ) ,  radical  a r e i n Ji(K)  . I  j = Y1TIr  of  i s a K-subring of  conclude  A e W(K)  J  N  class.  and the c a r d i n a l i t y o f  A  are i n  ^  M  So we  If 1  = S  suppose t h a t  A' fl B e M ; a n d s i n c e g  i s a radical  the c a r d i n a l i t y  Thus t h e c a r d i n a l i t y  A' £ K , so ' A' n B  _^A_  and  i s a K-subring of a r i n g  such that  Now both  we show t h a t  If  A c s  H * o  K ^  €  .  (*) and  M  ^) K  J b y  2  are  34(K)-ideals  of a r i n g  ' '  (  b y  I t follows  1  3  v i ) ;  a n d  that f i n i t e  s o  sums o f  R  «(K) - i d e a l s  H(K) - i d e a l s . As  dition  (**)  we have a l r e a d y  (A) b y 1.1.3  then  (vi).  noticed,  W(K)  satisfies  con-  -  Now .we s h a l l (B).  Let  R  of  R  ideals x  ty  .  W(K)  S  If  satisfies  i s a K-subring of  sum o f  W(K)  subring  of  i s a n ' tt(K) - i d e a l  property  and  fore,  is  I  J . S  proof  condition  and  of  of  R}  Now  tt  i s t h e sum o f tt - I d e a l s , (B)  and  class  34(K)  (**)  the c a r d i n a l i S n J  so  then  S =  R , so  .  S £ K  x e S  R , b u t by  S fl J e 3i , s o , s i n c e  Thus  tt(K)  The  I  - ideals of  S D J <_ t h e c a r d i n a l i t y o f  of  the  show t h a t  sum o f i d e a l s i s a n tt(K) - i d e a l n J : J  £{S  -  be a n y r i n g a n d l e t I = t h e sum o f a l l H ( K ) -  i s in a finite  this  16  i s a K-  is a  Sett.  radical There-  i s established.  satisfies  (C) b e c a u s e o f (*) so  i s complete.  Q.E.D.  2.2  LOCAL CLASSES. This classes. local S  Henceforth,  finitely  rings  Rett*.  R  is  with general  -classes w i l l write  tt*  tf -generated  properties  V\ ~ 0  be r e f e r r e d t o a s for  •  i f and o n l y  Q  of  A  i f S  subring is  g e n e r a t e d as a r i n g .  PROPOSITION: Let  class  K  c l a s s e s a n d we s h a l l  of a r i n g  2.2.1  s e c t i o n deals  tt  i f and o n l y  be a c l a s s o f r i n g s . i f tt* s a t i s f i e s  R , i f I O R  such t h a t  tt*  is a  condition  R/I e tt* a n d  radical  (A) a n d f o r a l l I e tt* t h e n  - 17 -  Proof: One way I s o b v i o u s s i n c e for a l l radical  satisfied  b y ft* .  of a l l '  ft*-ideals of I  ideals. finite  in  Let  of  then  R  S  R .  t h e two c o n d i t i o n s a r e  be a r i n g a n d l e t I If  S  i sa finitely  i s a subring  sums o f  2.1.4,  ft*-ideals  Therefore, ft*  ft*  are  on  ft*-ideals.  sum o f ft*-  ft*  imply  Hence  satisfies condition  satisfies condition  be t h e sum generated  of a f i n i t e  J u s t a s i n 2.1.4 o u r c o n d i t i o n s  l e f t * .  hold  classes.  C o n v e r s e l y , assume t h a t  subring  t h e two c o n d i t i o n s  ( C ) ; so  that  S e ft , s o  ( B ) . A g a i n as  ft*  i s a radical  class. Q.E.D.  The  following  t i o n f o r concluding This  that  theorem p r o v i d e s certain  c o n d i t i o n a n d t h e one g i v e n  when we c o n s i d e r  a sufficient  classes are radical i n 2.2.7 w i l l  s p e c i f i c l o c a l classes  condiclasses.  be u s e f u l  i n Chapter V.  2.2.2 THEOREM: If (i) (ii)  ft If in  ft  i s a class of rings  s a t i s f i e s condition A .is a ring, ft  then  satisfying:  (A) .  B <3 A  and b o t h  A e ft .  ( i i i ) ft* < ft . Then  ft*  i s a radical  class.  B  and  A/B  are  - 1 8 -  Proof: We s h a l l  show t h a t  the c o n d i t i o n s  of 2 . 2 . 1 are  satisfied. Since  ft  (A), by 2 . 1 . 3 ( v i ) ,  satisfies  ft*  satis  fies (A). Suppose t h a t A/B e ft* a n d ' subring Since  of  A  B e ft* .  A  .  Now  B e ft* a n d  so b y ( i i i )  A  A'  Let  such  A ' • be a f i n i t e l y s -  -—g-^  ft*  B <J A  i s a r i n g and  ?  A _ -  E  - M  generated  Therefore  A/B e ft* •  since  i s strongly hereditary,  0 B e ft .  by ( i i )  that  A  7  (1 B e ft*  A' e ft s o  e ft* . Q.E.D.  2.2.3  COROLLARY; I n t h e theorem, c o n d i t i o n  the  condition  that  (iii)  c a n be r e p l a c e d  the union of a countable  sequence o f f i n i t e l y  generated rings  in  ft*  by  increasing i s i n ft* .  Proof: In  the proof  A' n B e ft .  to i n s u r e  that  countable  i t i s clear  able  of 2 . 2 . 2  that  A  condition  PI B  since  ft*  and  A'  i  i s the union of a count-  i n c r e a s i n g sequence o f f i n i t e l y ft*  was n e e d e d  A' n B c A'  Since 7  (iii)  generated  These subrings  are  i s strongly  Thus t h e p r o o f  o f the c o r o l l a r y f o l l o w s  subrings. hereditary.  immediately. Q.E.D.  - 19  2.2.4  -  COROLLARY: If  equivalent (0) j£ A e  tt  i s a r a d i c a l c l a s s then c o n d i t i o n  t o the c o n d i t i o n tt*  then  tt(A)  /  that f o r a l l rings (0)  A  (iii) is , i f  .  Proof: Clearly r i n g and  condition  (0) ^ A e  tt*  ( i i i ) Implies that  then  W(A)  = A ^  (0)  i f  A  .  C o n v e r s e l y , assume t h a t f o r a l l n o n - z e r o if  A e  tt*  then  A/H(A) e M*  tt(A)  since  A/tt(A)  must be  ideal.  Therefore  ^  tt*  (0)  (0)  .  Let  satisfies  A e tt* .  condition  is a  rings  A,  Then  (A).  Thus  o r e l s e i t w o u l d have a n o n - z e r o  M-  A e tt . Q.E.D.  If  we  assume c o n d i t i o n s  problem  of showing  showing  that  such that 2.2.2  A'/B'  - 2.2.4  tt*.<_ tt . finitely  i f  we  A'  that  tt*  ( i ) and  B'  e  Another p o s s i b i l i t y  g e n e r a t e d r i n g and  tt*  have a c c o m p l i s h e d  g e n e r a t e d as a r i n g  the  i s a r a d i c a l c l a s s reduces  is a finitely  e tt* and  ( i i ) of 2.2.2  then  this  B'  e M  .  by r e q u i r i n g  i s t o show t h a t (and h e n c e i n  B'  tt).  B'O In that  must We  to  now  be turn  our a t t e n t i o n i n t h i s d i r e c t i o n .  2.2-5  DEFINITION: 3. £.  i s the c l a s s of a l l r i n g s  R  which  contain  A'  - 20 -  ,  f i n i t e s e t o f elements y e R  N y = Ea.x. i=l 1  on  x, ) NT  where the  a.  such t h a t f o r a i l  are integers  depending  1  y . The p r o o f o f the f o l l o w i n g lemma i s based on a  p r o o f g i v e n by Jacobson (page 2 4 1 ) [ 1 1 ] .  2.2.6 LEMMA: If such t h a t r i n g of  A  B «3 A  i s a f i n i t e l y g e n e r a t e d r i n g and  A/B e  .  then  B  i s a f i n i t e l y g e n e r a t e d sub-  A .  Proof: .Choose y e A/B  x^, • .., x^  in A  T  there are integers  a^, .. .  Ci-. X-, + .. . + ct^x^ and such t h a t set  of generators of  b. . e B  ij  the  such t h a t  subring of  B  A .  f o r which  {  }  Select integers N T,  x.x. -  i J  such t h a t f o r a l l  r  T  ijK K  K = 1  T  i j  T  cases i n which  N . b  Let  _ B  be  where a l l the s u b s c r i p t s v a r y  F i r s t we s h a l l show t h a t i f b e B M = 1,  and elements  g e n e r a t e d by the f i n i t e s e t Y =  fb. ., x^b. b. . x , x^b. .x } IJ J\ I j IJ L is. I J L from 1 t o N .  for  contains a  Y±jYi  y. ,. x , + b. . .  y =  then  x b e B M  C l e a r l y i t s u f f i c e s to consider the  i s a generator of  b. .. the r e s u l t - i s o b v i o u s . ij  B .  Suppose  I f b = b. .x b = x b. . L 1j  T  or  Then  -  -  21  N X  M  b  =  ( x  M L X  ) b  ij  _  ^f^M^K  =  X  K  +  b  e B  ML) lj b  .  And I f  N = L ij H  b  X  b  X  (b .x ) l j  M =  t  e B  H  h  e  n  .  X  M  -  b  ( x  M L X  Similarily,  X  (  =  i f b e B  f M,L,K Y  K  x  then  bx  M  1=1  1  ( x ^ , . .., x )  contains  N  s u f f i c i e n t to consider  1  0  K  all  +  b  € B  ML'  for  a s e t of generators of  t h e c a s e when  a = x.  .  A  i t  ... x. K X  —  our d e f i n i t i o n <_ 2  K  1  1  By  X  N e x t we s h a l l show t h a t i f a e A there e x i s t N _ n. such that a = E n.x. + b where b € B .  integers  is  ^iJ H)  ..., N .  1,  Since  ) (  Let  K < M  of  M > 2  .  Now,  B  i t i s clear  that  and s u p p o s e t h a t  this  i s true  the r e s u l t  when  holds f o r  i f  a = x. ... x. = ( x . ... x. )x. ^-l ^ i ^ - 1 M N _ N ( E n.x. + b ) x . where b e B , then""a = E n.x.x. + j=l l l i j ji N N N bx. = E E n.y^ x + E b .. + bx. and t h i s i s o f M j = l K=l ^ M * j=l M M x  M  J s = 1  K  required  J 1  form since  Now  integers]  additive  group  C A  generated a b e l i a n additive for  group.  e B .  N C = ( E n.x.  consider  .  X  bx.  1=1  are  M  K  J  the  X  1  which a r e i n  Since  group, Let  C  (c^,  C  : n. 1  i s a subgroup o f the f i n i t e l y .  B  1  i s a subgroup o f a  is finitely c^}  generated finitely  g e n e r a t e d as a n  be a s e t o f g e n e r a t o r s  C . We now show t h a t  X = Y U (c-^,  c ) M  generates  - 22 -  B  as a s u b r i n g  z e B  .  Then  of  A  z e A  .  Since  so  z =  X c B  N " £ n.x. + b  1=1 and  the  n^  1  1=1  1  so  B = <X> .  and  .  Suppose _ b e B c <X>  where  1  z - b e C .  a r e i n t e g e r s ; hence  N S n.x. 6 C c <X>  , <X> c B  _ b e B c <X>  .  That i s ,  Therefore,  B e  <X>  1  Q.E.D.  2.2.7  THEOREM: If  (i) (ii)  ft  i s a class of rings  satisfies  A , i f B <3 A  a r e i n ft t h e n  If  A  such  that:  such  that both  condition (A),  For a l l rings B  (iii)  H  A/B  and  A e ft .  i sa finitely  generated  ring  A e ft t h e n  and  A e 3. £>. Then  ft*  i s a radical  class.  Proof: We s h a l l  show t h a t  the conditions  By 2 . 1 . 3  ( v i ) , ft* s a t i s f i e s  of 2 . 2 . 1 are  satisfied.  Suppose ft* .  Let  -—g-^- = 2.2.6  A  A' i  that  B <£3 A  be a f i n i t e l y  and both generated  ^ g e ft a n d b y ( i i i )  implies  that  A' fl B  &'^'r)  i sfinitely  condition (A). A/B subring  and  B  of  A  g e ^-S-  are i n .  Then  Lemma  generated as a  subring  - 23 -  of  B , so s i n c e  so  A  B e 34* , A' n B e 34 . Now by ( i i ) A' e «  i s 34* .  T h i s completes t h e p r o o f . Q.E.D.  2.2.8  COROLLARY: 34 i s a r a d i c a l c l a s s and  If 34*  i sa radical  t o be a r a d i c a l  2.2.9  s u f f i c i e n t conditions f o r  class.  THEOREM: If  7$, i s a c l a s s o f r i n g s s a t i s f y i n g c o n d i t i o n (E)  Tfi a l s o s a t i s f i e s t h e c o n d i t i o n t h a t i f R  and  then  class.  The next theorem p r o v i d e s (U^)*  34* <_ 3 . B.  i sa finitely  generated r i n g i n "YYl then every non-zero homomorphic image of R  can be homomorphically-mapped onto a non-zero r i n g I n 77? ,  then  (*W»y^')*  i s  a  radical  class.  Proof: Clearly  (H^)*  Suppose t h a t both  A  and  A/B  I  o f A'  s h a l l consider  B  are i n  generated subring of ideal  s a t i s f i e s condition (A). i s an i d e a l of a r i n g (U )* M  . L e t A'  A  and t h a t  be a f i n i t e l y  A . I f A' k l / . - ^ then t h e r e i s an  such t h a t  A'/I / ( 0 )  and  A'/I  e YA .  two p o s s i b l e cases and show t h a t t h e y b o t h  lead to a contradiction.  We  - 24  Case 1:  A'  In  t h i s case  is  strongly  A / I =  CU-**)*  B  o I  finitely  •  generated.  In  t h i s case  onto  fl B  A'  T  h  u  A  =  ring  '/  1  €  j  n  •  s  i  n  c  bmce  n  , A'  ^>*  B  e  e  KU.^)  C&jvJ* •  ring A  /  /  I  i  s  contradiction.  .  A' —p—— A' A fl B + I c  a  n  b  e  ;  a l s o a homomorphic image o f  B  (A), the f a c t o r  is a  + I / A'  ^  (U.^)*  A e  s  This  A ' 4- B g-—  the non-zero  and  =  satisfies  e  Case 2:  . j  hereditary  Then s i n c e A'^n  D B + I = A'  -  A'/I  h o m o m o r p h i c a l l y mapped A' ' A' D B + I S  e  I  N  C  i  E  ; by  s  assumption,  A' ——^  -g  ring  i n 7&€  1  - ' j . c a n be .  But  homomorphically  this i s a contradiction  Since both that  A'  radical  e Uyt  •  c l a s s by  mapped o n t o a  cases l e a d  Therefore,  since  non-zero ^  ^ ^  t o a c o n t r a d i c t i o n we  A €  and  e ]/^.  conclude  so  is a  2.2.1. Q . E . D .  2.2.10 COROLLARY: If is  YVL  satisfies  a radical class.  simple r i n g s  then  condition  ( E ) and  I n p a r t i c u l a r , i f yvt  (tC^)*  (U^)*  (A) t h e n  i s a class  i s a radical class.  of  -  2.2.11  is  34  i sa radical  image o f a n  a radical  34 s . s .  ring  class  a n d no n o n - z e r o  isin  14  then  homomor-  34* = (U-u  0  0  )*  class.  The  next  lemma w i l l  2.1.4 i s n o t t r u e when lower r a d i c a l by  -  COROLLARY: If'  phic  25  the class  class,  K = ^  be u s e f u l .  Q  i n showing  Recall  i s the lower r a d i c a l  of a l l nilpotent  that  that  /3 , t h e B a e r  class  determined  rings.  2.2.12 LEMMA: For-any r i n g (i)  (ii)  i f  (0) / I  is  an i d e a l  I®  c J  i f  i s an a c c e s s i b l e J  of  R  finitely  s u b r i n g of  generated as a s u b r i n g o f  of R  R  then  there  and  N .  generated as a s u b r i n g o f J  of  (0) / J c I  integer  i s an a c c e s s i b l e  a non-zero i d e a l  then,  subring  such that  f o r some p o s i t i v e  (0) / I  exists  R , i f ]3(R) = (0)  R  R  R  then  which i s there  which i s f i n i t e l y  and such t h a t  J _ I .  Proof: (i)  We w i l l  prove  degree o f R  .  Suppose t h a t less  I  .  <1 R  result  I'. If  degree N  this  than  Now  by i n d u c t i o n  N = 1 , I  the r e s u l t N .  ((I)j )  Let  itself  on  N = the  i s an i d e a l o f  holds f o r a l l subrings of I = I  1  <J I  2  <3 I ^ <  c I . by A n d r u n a k i e v i c ' s  ••• O Lemma  - 26 -  (page 1 0 9 , ((I)  )  T  1  [7])-  j3(R) = ( 0 )  Moreover, s i n c e  i s an a c c e s s i b l e s u b r i n g  of  R  [7]  ( ( I )  that  1  by o u r i n d u c t i o n J £  (ii)'  ((I)! ) .  Then  hypothesis,  and  3  J c I  (((l)j and  (0) /  and  is finitely  that .  I  and  {z^,  I  c J  N  z^-}  1  we  3  ^c J  3  3  4 (0) .  Thus,  3 (0) 4 <  exists  £ J  integer  so we a r e d o n e . of  R  (0) / J « 3 R  : L <_ N-1)  such  that  N . Let  i s a finite  I  .  Then  s e t such  that  L I / J then there  2.2.6  is a finitely  J  completes  .  are integers  Therefore generated  o^  such  I / J e 3. B . subring  that so by  of  R .  the proof. Q.E.D.  We now p r e s e n t  2.2.13  ft*  a n example  i s not a r a d i c a l  of a r a d i c a l  class  ft  class.  EXAMPLE: Let  commuting  R,  J  f o r some  f o r some i n t e g e r  : d e D}  such that  from  generated as a s u b r i n g o f  exists  w = Efo^d  This  ) )  )  be a s e t o f g e n e r a t o r s o f  D = (z. ,...z. 1  there  T  i s an a c c e s s i b l e s u b r i n g  T h e n by ( i ) t h e r e  J c I  if  I  I  Assume t h a t  R  i tfollows  3  Lemma 33 i n D i v i n s k y  N  and  K  be t h e f r e e r i n g g e n e r a t e d b y two n o n -  indeterminants Define:  M =  x (2y)  and K  y .  -  A =  27  K/M  B = (2x) =  LEMMA  A  {I : I  i s an a c c e s s i b l e  subring  of.  A}.  A: I eJH  If only  -  i f  I c B  ,  then  I  i s commutative  i f and  .  Proof: First 2yx  e M  2x-y  2xy  and  = 0  we  show t h a t  B  i s commutative.  , y-2x = 2yx = 0  e M  in  A  in  where t h e  A  .  Hence,  i f  a^  b e B , b =  a r e even i n t e g e r s  such a r e p r e s e n t a t i o n  then,  B  and _.  N  E i=l  that  Since  of  d e p e n d i n g on  b  i s unique).  a.  x  1  b  (notice  Clearly  i s commutative. To show t h e c o n v e r s e we  b e g i n w i t h the  ing  c a l c u l a t i o n s , t h e p u r p o s e o f w h i c h i s t o show  if  u £ B  0 4 u € A  then  are  distinct  of  x  and  x u / ux  and  b e B  where  u £ B  = 0 = by  yu ^ uy  Then  , the  u = c^m^  ( b y a monomial  with c o e f f i c i e n t  we  x(m^  +  ... + m^)  y(m,  +  ...  have  .  = 1 that  1).  .  Since  xu  - ux =  - (m^ +  ... + m^)x  + m r r ) - (nu +  ... + m )y Tr  we  that  ... + %i\  + b  a  and t h e  nu  mean a p r o d u c t  M o r e o v e r , we xb  and .  follow-  Suppose +  a r e odd i n t e g e r s  monomials y  or  .  assume t h a t a l l t h e yh  1  If  may  - bx = 0  and  yu - uy = u = x  or  - 2 8 -  y  or  x + y  then since  O t h e r w i s e we  may  assume  necessary) that e Ky xm-^  .  Now  .. . + xm^  Since  A e M =  the  n^  are  ( 2 y )  l  +  m  then  j  =  0  m^  w  i  c  h  i  is free.  be  e q u a l t o one  of  (*•).  an  e v e n number o f  But  .  Thus  Both of  cases a l r e a d y  xra o  xm,  (*),  the  and  since  n . = xm.  similarily  any  case e i t h e r Now  and  that  I  xu  suppose  and  / 0  I  i s an  i s commutative'.  xm^  the j  n .  that  xu  •'  4 0  or  / 0  yu  =  5/ 0  are  m^x  .  since  xm^  must  r i g h t hand However,  a p p e a r on for or  n.  t o one  .  yu  I cf: B u  e I  .  Now  j  of  some  the con-  assume . 0  Thus .  subring  t h e n by , u yu  right we  to a  4 0  - uy  since  = m.x  I f we - uy  side  the  i 4  accessible  exists a - uy  yu  or  If  b^  assumption  i / 1  - ux  (*)  where  i  the  is free,  - ux  .  f o r i f so,  our  show t h a t  c a l c u l a t i o n s there  A e M  both of which l e a d xu  above  - ux  K  t h e s e s i t u a t i o n s l e a d us considered,  Suppose  1  n . 4 n.  for  if  i  where  If  f o r some  monomials  .  m  A = Eb n  4 0  K  the  m^ in  since  finished.  the  A  write  + xm^.  1  = n.  Therefore,  , we  can  +  monomials on  tradiction. e Kx  + m^-x  permitted.  t  are  have  monomials i n  again, of  must have t h a t i  n  we  which c o n t r a d i c t s  K  of  s  we  e Ky  = 0  ...  , we  K  Now,  € Kx  hand s i d e  +  distinct  h  or  - ux  = m^x  even i n t e g e r s . m  xu  4 yx  (by r e a r r a n g i n g  e Kx i f  +  xy  £ B  of the  , such  e l  for  A  - 29 -  some p o s i t i v e I n t e g e r (yu u  - uy)u  N  = 0 .  N  so  Thus  y u -u - u - y u  (yu - u y ) u  | B , u | 2K ; t h e r e f o r e ,  Since  y u - uy  yu  - uy € ( 2 y )  xu  - ux ^ 0 .  i s also i n = M .  K  xu  only  - ux = 0 .  (y)j^  Since  x  e 2K . . S i n c e  N  £ 2K  N  so  y u - u y e 2K.  i t follows  yu - uy = 0  As a b o v e i t f o l l o w s  However, s i n c e involves  u  =  that  that  we must have  x u -. ux e 2K .  commutes w i t h a n y monomial i n u  x ,  x u - ux e ( y )  This  i s a c o n t r a d i c t i o n so  K  •  We c o n c l u d e  that  I £ B . Q.E.D.  LEMMA B: No contained  non-zero a c c e s s i b l e  in B  i sfinitely  subring  of  A  which i s  generated as a s u b r i n g o f  A .  Proof: By  Lemma 2 . 2 . 1 2 ( i i )  sider  t h e c a s e when  I O A  , I c B  seen that  and  the  coefficients.  b  I<  A  and  I c B  I = <b ,  L e t L = maxfdegree  suppose t h e degree o f  a^x  + ... + a-jOc^  b  where t h e  w = max{n : 2  n  divides  Suppose We have  k  must be p o l y n o m i a l s  i  .  t o con-  b > / (0) .  1  and  Let  i t i s sufficient  H  a  Now,  a r e even  i  x  with  = integers.  a l l i = 1,  , L}.  wL Define  b = x  b„ e I .  even  b^ : i = l,...,k}  = L . a^  in  The s m a l l e s t  power o f  x  - 30 -  which appears w i t h a non-zero wL + 1 . of  Since  b e <b^, . b  products o f the  b ^ <_ L  for a l l  a non-zero  coefficient  b^'s .  >  K  , b  Because  of  b  b  is  must be a sum  the degree o f  i , each p r o d u c t which  coefficient  in  contributes to  must c o n t a i n a t l e a s t  w+1 w + 1  terms  efficients  of  coefficients b  H  a n d hence b .  of  2  This  must d i v i d e a l l t h e c o i s a c o n t r a d i c t i o n because the  b  are exactly  the c o e f f i c i e n t s of  I  c a n n o t be f i n i t e l y  . Hence  generated. Q.E.D.  LEMMA C: No in  accessible  the i d e a l  of  (y)  A  non-zero  i sfinitely  subring of  A  contained  generated as a s u b r i n g  A .  Proof: We c a n a p p l y Lemma 2 . 2 . 1 2 ( i i ) a g a i n s o we need  only  consider ideals Suppose t h a t  I  = <z.., .... z,,>  m.-. + ... + m. ll iK  h  a l l the  such that  A .  'I<3 A ,  where e a c h  (0) 4 I c ty)  z. 4 0  and  f o r n o n - z e r o monomials  v  sider  of  ±  m^ .  some  and l e t m.^  e x K h  and  z. =  m. . . C o n i j  d  be t h e l a r g e s t  .  I t follows  integer  that i f  - 31 -  w e <z^, w e I This  ...z^>  and  w / 0  then  max{h  : w e x K}  then  x^ w  e I  + 1  i s a contradiction  so  g e n e r a t e d as a s u b r i n g o f  <_ d .  h  I  and  cannot  x  be  d + 1  Buti f  w 4 0 .  finitely  A . Q.E.D.  We a r e now r e a d y  to prove:  THEOREM:  IX-M.  is a radical  class  (Vb^)*  but  i s not a  Pi  radical  class.  Proof: The is  a radical  class class  ^  R e IXy^  and  n o t be h o m o m o r p h i c a l l y  property (E)'so  }JL^  i f and o n l y i f R  can  satisfies  mapped o n t o a n o n - z e r o  A  ring i n  "ftt . In bc u a lt  cA l a s{s  o r d e r t o show t h a t  (UJ* we w i l l. Since  it  i s clear  a finitely  then  R'  A  that  To is  show t h a t  A/B e (]L )*  m  see t h a t generated  g e n e r a t e d and  A/B e ( t i ^ ) * subring of  I ^ (0) .  radi(U*)*  A € TVl  .  c a n be h o m o m o r p h i c a l l y  R'/L =" I €*)tyt a n d  , B e  M  is finitely A £ (U )*  i s not a  , suppose t h a t  , R'  A/B  ]yC^  mapped  Since  A/B  .  If  R'  onto i s a ring of  -  characteristic Clearly  (y)  2 ,  I  3 2 -  must be o f c h a r a c t e r i s t i c 2 .  contains  A  a l l accessible  which a r e o f c h a r a c t e r i s t i c 2 so is  a homomorphic  image o f  g e n e r a t e d as a r i n g . fore/  R' e \  This  tative  generated  so i f R  and  finitely  and  b y Lemma B t h i s  subring  •  K  of  B  , suppose  .  generated as a r i n g .  Hence  This  Since  I  must be f i n i t e l y  (14*^)*  I e t/i a n d  implies  generated as a r i n g . e U  A  There-  Then  R'  R  i sa  i s commu-  7  i t c a n be h o m o m o r p h i c a l l y map-  onto a non-zero  7  , I  7  •  A  c o n t r a d i c t s Lemma C.  B e  7  ped  R  I c (y)  of  A/Be.(U^*-  so  To s e e t h a t finitely  R  subrings  This  that  must be commutative  B y Lemma A , I c B I  i s not f i n i t e l y  i s a c o n t r a d i c t i o n , so  B e  completes  I  . the proof. Q.E.D.  Notice two e l e m e n t s , 2  1/  £ < H K  (K)  that  2.3  0  *  that  since  A £ ]/i (K) T h u s  >  i s generated by only  f o r any c a r d i n a l  although  i s not a r a d i c a l  2 £ K <_ ^  A  K  such  that  l l ^ i s a radical class,  c l a s s f o r any c a r d i n a l  K  such  classes  J*(K)  • .  ELEMENTARY CLASSES. The p r o b l e m s i n v o l v e d where  2 < K <  ^£  In dealing with  a r e s i m i l a r t o those i n v o l v e d  i n dealing  - 33  with l o c a l c l a s s e s .  For  -  t h i s reason we  f i c a l l y w i t h such K - c l a s s e s but t i e s of 2 - c l a s s e s . to 2 - c l a s s e s  We  A subring  for  i s a 2 - s u b r i n g i f and element.  homomorphic images of the r i n g integers  tt'  i n one  tt(2)  $[X]  variable  X  and  w i t h zero constant  ®X]/I  where  s e t of elements  I =  (f (X),  element f (X))  ] L  (f (X),  R  are for  R  f (X)) £  1  (JtX] .  p r o o f s of the f o l l o w i n g three r e s u l t s are  s i m i l a r to the" p r o o f s of 2 . 2 . 1 ,  2.2.2  and  2.2.9  respectively.  PROPOSITION: Let  c l a s s i f and rings  refer  of a l l p o l y n o m i a l s  of the form  2.3.1  and  Such r i n g s are a l l  Hence a l l r i n g s generated by one  The  proper-  only i f i t i s  coefficient.  some f i n i t e  speci-  classes.  generated, as a r i n g , by one  over the  deal  w i l l pass on to the  s h a l l write  as elementary  w i l l hot  R  tt  be a c l a s s of r i n g s .  only i f  , if  K R  tt'  satisfies  such t h a t  R/I  tt-'  is a radical  condition  (A) and  € tt' and  for a l l  I e tt' then  Rett'.  2.3.2  THEOREM: If  (i) (ii)  (iii)  tt  tt  i s a. c l a s s of r i n g s s a t i s f y i n g :  satisfies  condition  If  A  i s a ring,  in  tt  then  tt'  <  K  .  (A),  B <Q A  A e tt .  and  both  B  and  A/B  are  - 34 -  Then  34'  i s a radical  2.3-3  THEOREM: L e t 7A  (E).  be  class.  a class  I f a l l non-zero  satisfying  condition  homomorphic images o f r i n g s  w h i c h a r e g e n e r a t e d by mapped o n t o n o n - z e r o  of r i n g s  one  element  rings  "YA  in  c a n be  homomorphically  (tl^)'  , then  YYl  in  is a radical  class.  In classes. (I*a)  =  [13]  If  Rjabuhin discusses is  A  (f(X) e  I o  a ring,  <?(X]  : f ( a ) € 1}  of  fJ[X]  satisfying:  (i)  i f  A o B  e R  (ii)  i f  B  (iii)  i f  A 3B  ideals  e R  then  and  A <  and  and .  f$X]  such a s e t o f i d e a l s  f o r any  R , the c l a s s  R  such that  is  an r - s e t  buhin and is  calls  that  i f  an r - s e t  line  of the  (((0)*a) then  define  be a n y  A e R  s e t of  .  (B*f) e R  then  .  , (B*f) e R  f(X) e A  .  Rjabuhin c a l l s r-set  R  then  and f o r a l l  radical  a e A  Let  f ( X ) e ' (p[X]  , A e R e R  B  A  elementary  is  such classes  £ R  .  ft(R) He  an e l e m e n t a r y  and  is  R  such that  proves  that  .  i f  r a d i c a l class  elementary r a d i c a l class 34 = 34(R)  defines  t o be a l l r i n g s  semi-strictly hereditary  34  any  of rings  : a e R}  5t(R)  an r - s e t ;  We  (Rja-  radicals)  then  there  s h a l l g i v e an  proof.  Suppose t h a t  R  i s an r - s e t ,  then  R  ( i ) implies  out-  ,  - 35  that  tt(ft)  and  B  are  ((0)*a) that  has  property  in  tt(ft)  ((0)*a)  e  tt(ft)  .  Let  By  the  very  that  P[X]/1 e  Notice  f(X)  .: P [ X ] / I e  A/B  ( i ) ; and  (ii).  e A e  Hence  B  so  r - s e t and  2.3-4  since  SO clearly  e  ((0)*a)  .  Now  ((0)*a)  £  A/B  (B*a)  (iii)  =  implies  , then  f(a)  e B  tt(ft)  ,  .  Thus  tt(ft)  2.3.1,  ( i i ) implies  both  is  tt(ft)  that i f  an  is a  l e f t  then  tt' .  i s a r a d i c a l c l a s s l e t jaf =  Since  tt'  This  tt'  tt'  satisfies  ,J  (A)  i s s t r o n g l y h e r e d i t a r y , <af and  A  implies  e  dl  and  <_?  satisfies  tt'  = tt(<J ) .  for a l l  that  e  i s a r a d i c a l property, (iii).  sa-  tt'  and  (p['X3/B e tt'.  Therefore  (J  is  an  THEOREM: Let  tt'  be  an  c l a s s and  which contains  R  s.s.  is  A/B  Proposition  A 3 B  e ft .  since  e  tt'3  Suppose  , (B*f)  tt'  in  and  tt(ft) .  Conversely, i f  tisfies  , then  d e f i n i t i o n of  radical  class.  BOA  e A  ( ( 0 ) * f ( a ) ) e ft s i n c e  our  satisfies  a  f(X)  e l e m e n t a r y c l a s s so by  ( I <J (p[X]  Suppose  a" = a + B  e ft f o r i f  (((0)*a)*f) = A  (A).  .  € ft where  -  tt'  direct radical  sum  i f and  of prime  class.  tt'  elementary c l a s s which i s a r a d i c a l the  only s.s.  Baer lower r a d i c a l if  R  rings.  i s isomorphic Hence  tt'  /3 . to a  A  ring  sub-  is a special  - 36 -  Proof: S u b d i r e c t sums o f s e m i - s i m p l e r i n g s a r e s e m i - s i m p l e s o we n e e d o n l y show t h a t i f then  R  rings. x  Is  x  I  ft'  and  x  Suppose  N  u  a  Then  D <y>  Since  <y>+N N  N  £  + I  ft'} is  s.s.  Then  of  R  element  such  that  ring. 0 / x e R .  Z / 0  Then t h e r e  <y> £ ft' .  since  there  for a l l y  a  (0) e Z .  y e A  a .e--A • . " " ' L e t  so  Therefore  is  •  N e Z .  <  ^  s  in  such  N = U{N  ^  f  is  Let  i s an a s c e n d i n g c h a i n o f i d e a l s  has A . C . C .  Y  ..  .  I  and hence  W  ft's.s.  an a s c e n d i n g c h a i n of i d e a l s  n< > = N n< > y  and  <y>  : a e A <y>  ft's.s.  ft's.s.  that  n <y>  ft'  ft'  f o r each non-zero  R , an I d e a l  be  h  ^  < y  n <y> c  a  c  to f i n d ,  a prime  : a e A  N .  is R  s  Z = ( I <J R :  <y>  ring  R/I  y e ( )R  Then  sufficient  s.s.  Let a  is  i s o m o r p h i c t o a s u b d i r e c t sum o f p r i m e  So i t i s  of an  R  always  =  ^  a  Z . In  that : a e A}.  _  Thus we may c h o o s e ,  by  Y  Z o r n ' s Lemma,  I  Since  y £ I  To. s e e  that  t h e n c e Thus  maximal i n  J / I £ ft  <y>HJ+I c ^ .  Z .  and R/I  y e (x) is  ft's.s.,  B u t ^ . M ' 7  since  ft'  Therefore  is R/I  , x  I  .  suppose  so  that  Z  s t r o n g l y h e r e d i t a r y and is  ft's.s-.  .  J< R  M ' •  - 37 -  N e x t we s h a l l and  Jg  J  then  n J  1  subdirect  are Ideals 2  p I  sum o f  whi ch a r e i n tt' ; and  Rg  and <  y  +  e tt' .  I  J, n then  R/I  R  1  R  J  R/I  Is a prime r i n g .  which p r o p e r l y  For i f ^ <y>+J-, = — j  0  ^  and  = 1  tt'  X  2  £ I  then  <  y  ^  +  If  I ' isa  I  2  both of  d i r e c t sum o f  i s strongly  sum i s a s u b r i n g J *J  contain  <y>+J Rg .= — j  hence, the ( e x t e r n a l )  Now i f  R^  hereditary  o f t h e d i r e c t sum,  then  n J g ) c I . But  (^  2  •  -——:  clusion  .  of  i s i n tt' a n d s i n c e  the s u b d i r e c t  x  show t h a t  € j3 <_ tt w h i c h c o n t r a d i c t s  that  R/I  is  tt's.s.  our previous  Therefore,  J-^-Jg £ I  con-  so  i s a prime r i n g . This  completes  the proof. Q.E.D.  I n Theorem 2.1.4 we p r o v e d class  and  tt(K) i s  K  class  class  class. tt  classes  analogous  i s unsolved.  K  tt(K)  such that  question  isa  K ^  radical  K  then  was n o t a r a d i c a l 2 £ K <_  concerning  .  elementary  T h a t i s , we do n o t know w h e t h e r o r n o t  tt' = tt(2) must be a r a d i c a l class.  M  I n 2.2.13 we p r e s e n t e d a n example  such that  f o ra l l cardinals The  i f  i s a c a r d i n a l number s u c h t h a t  a radical  of a r a d i c a l  that  c l a s s whenever  tt  isa  radical  - 38 -  2.4  GENERALIZED K-CLASSES: In with The  the past  the question,  three  sections  we have b e e n  "When a r e K - c l a s s e s  radical  purpose o f t h i s s e c t i o n i s t o consider  "When a r e K - c l a s s e s  semi-simple  classes?"  c r i b i n g a c l a s s o f r a d i c a l s which contains where  2.4.1  34  i s a semi-simple  classes?"  the question, We b e g i n b y d e s a l l r a d i c a l s t/jj  K-class.  DEFINITION: Let  34 be a c l a s s o f r i n g s  number w h i c h i s (i)  34 \ g(K)  and  K  >_ 2 : i s the c l a s s of a l l r i n g s  fir  a cardinal  R  e v e r y n o n - z e r o homomorphic image o f a non-zero K-subring i n (ii)  J  i s a generalized  radical  i s a semi-simple  simple r i n g s .  is  we s h a l l  prove  a generalized  >_ 2  34  The r a d i c a l c l a s s  c l a s s determined by  For K  such that  that  class  contains  <_f  3" =  i f  then there  i s some #f  i s the c l a s s of a l l  34 ; t h a t  i s exactly  semi-  the upper  i s , <J =  .  i f 34 i s a l s o a K - c l a s s  In 1/.^  then  K-class.  any c l a s s o f r i n g s  we may f o r m t h e c l a s s e s  ft(K) i s  that  34 .  &  $$  R  i f and o n l y  If class  radical  K-class  f o r some c l a s s o f r i n g s  34  such  34 .  34 , s g(K) v  2.3'4  concerned  always a K - c l a s s  ft &(K)  a n d a n y c a r d i n a l number and  R  g  (x)  '  T  h  e  c  b u t need n o t be a r a d i c a l c l a s s  l  a  s  s  - 39 -  ( e v e n when  ft  is itself  three'sections  a radical class).  The p r e c e d i n g  o f t h i s c h a p t e r have b e e n l a r g e l y  with conditions  on  R  which  imply that  ft(K)  concerned  i s a radical  class. As we s h a l l p r o v e b e l o w , class. not  However t h e c l a s s o f  be a K - c l a s s .  conditions  on  semi-simple  R  which is a  the f o l l o w i n g  between K - c l a s s e s  2 . 4 . 2 ( i x ) we p r o v e  s  that  a  l  w  a  v  s  radical  a  rings  need  be c o n c e r n e d  the c l a s s o f  R  we l i s t  basic  that  K-classes  and p o i n t  and g e n e r a l i z e d  g  some  o u t some  i f ft i s a g e n e r a l i z e d  relaIn  K-class  K' >_ 2  then  .  PROPOSITION: ft  and  R  c a r d i n a l number w h i c h  be c l a s s e s >_ 2  are  of rings  and  K  and  .  ( i ) ft / T ^ N i s a r a d i c a l c l a s s .  \ ' (ii) (iii)  g(K)  I f ft < R If K < T  then  < ft„tv\ • g(K) - g(K) t h e n ft s < ft / „ v . "A„i \ v  (ir  g(K) - g ( r )  ( i v )  ( w  (v) ( V 1 )  with,  (K)  K-classes.  a r a d i c a l c l a s s f o r any c a r d i n a l  Let be  i  semi-simple  proposition  tionships  2.4.2  (K)  K-class.  of generalized  is  (K)  guarantee  properties  ft(K')  g  g  Much o f t h i s s e c t i o n w i l l  rings  In  R  a  .  g(K) g(r) )  I f ft < R < ( W  ± *g(K) • ft  g(K)  g ( K ) ) g ( K ) - *g(K)  for  then l  f  a  some c l a s s o f r i n g s  R n  g ( K )  ° ^  d  3  = i  f  which  ft  g(K)  .  *g(K)  =  'g(K)  satisfies (A).  r  - 40 -  (vii)  g(K)  = (»( ))g( ) 1 g ( 2 )  g ( K )  =»  (*(r)) (»(r))  r  M  g ( 2 )  (3* /x,N)(r)  < (» ,„»)  \  \ g(K)  -  (ix)  ; v  (M ( ))(r) g  l  * < ( ) >  f  M  g(K)'g(2) - g(K)  = (( g( ))g(2) M  K  r  •  (viii)  y  a n d  2  )  (  r  l  )  s  a  r  a  d  i  c  a  class.  l  K  Proof: (i) Prom t h e d e f i n i t i o n i t i s c l e a r t h a t (K) satisfies c o n d i t i o n ( A ) . We w i l l show t h a t 3i s also satisw  g  (ir  fies  c o n d i t i o n (D).  homomorphic  image o f a r i n g  ^g(K)"ideal. of  R  .  Let  Suppose t h a t  Therefore  contains a non-zero  M \-ideal g(K)  of  f v  which i s i n  ^g(K) 3J <_ R  n o n - z e r o homomorphic a hon-zero  R  be a n o n - z e r o homomorphic  7  i s also a K-subring  Now, b y 1 . 1 . 2 , (ii)  R  Then t h e  zero K-subring ring  Suppose t h a t e v e r y n o n - z e r o  ^  s  M .  of a  and  R  contains a non-  Of c o u r s e , .  7  this sub-  Therefore  radical  R e ^ (K)• g  class.  R e *g(£) •  image o f  K-subring  R'  image  R .  which i s i n  L  e  t  Then W  '  R  R  be a contains  7  a n d hence i n  R .  R e R ,v . g( ) /t  K  (iii)  Suppose t h a t  and  R e  M g  (  n o n - z e r o homomorphic  image o f  a non-zero K-subring  which i s i n  this (iv)  K <T  subring i s a T-subring  Suppose  homomorphic T-subring  image o f S c R  7  R  .  l  )•  R .  so  a n d  R e (Wg(K)^g(r)  K  Let  Then  W .  R  '  b e  be a contains  7  K <_ T ,  •  K  t  R  7  Since  R e g(r) e  R  a  n o n  ~  z e r o  Then t h e r e i s a n o n - z e r o  such that  S e ^(K)  •  Since  S  - 41 -  is  a n o n - z e r o homomorphic  image o f i t s e l f  S  t a i n s a n o n - z e r o K - s u b r i n g w h i c h i s i n ft . R  (v)  Suppose ft < ft <  a  n  ft  •  g(K)  <  d  i V  ^g(K) — ^ g ( K ) — ^g(K) Assume t h a t satisfies  (  g  zero  homomorphic  zero  K-subring  *g(K)  =  ( W  Since  g(K)  A  d  S  O  r  s  K  = ^„r \ g(K)  R e ** (K)  a n c  g  of  R .  R'  T  h  e  n  2  contains  h  s  l  n  c  (  K  )  )  g  (  S  e  M  *  S  (K)  satisfies R  '  b  e  a  n  condio  n  ~  i s a non-  S e JT .  Since  i s a K-subring,  R e ( » K  S  Conversely,  Then t h e r e  (A) a n d  g  u  g  such that  b e f o r e  < (ft  T  <  g(K)  g ( K ) = *g(K) '  where  v  image o f S  :  g  •  Suppose t h a t  g  (  K  )  )  g  (  K  •  )  soby(iv),  )  R e  be a n o n - z e r o homomorphic a non-zero K-subring  Hence e v e r y 2 - s u b r i n g o f in  R  ft  g ( K )"  M  K >. 2 , b y ( i i i ) we have t h a t  g  is  g(K))g(K)  n  <  g(R)  g(K))g(K) '  ( ( )) ( ) R'  '  condition  ft  ( J t  ft  (A) we a r e f i n i s h e d .  * g ( K ) = *g(K) '  Therefore  K  g  Suppose  satisfies 6  K  f x r  (A).  )  By ( i i ) ,  ( ) ) ( ) = * (K) '  g(K)  tion  S  (vii)  w  condition  assume t h a t ft  3  Hence  *g(K) '  £  <*g(K)>g(K)  (vi)  con-  ft(r)  R'  n ft s o c e r t a i n l y I f ft  < ft(r)  (*K )) ( ) r  (»(r))  ( )  a  K  image o f S  g  R  nd let  .  such that  Then  R  S e  ft(T).  which i s contained R e (*( )) ( ) r  .then by ( i i )  g  2  ft  g(2)  >.  K  n  in  7  S  *g(2)'  <(«(r))  g ( 2 )  - 42  so (viii)  («(r))  = »  g ( K )  Suppose t h a t zero  R  subring  S  S  W  €  r  By  n  d  .  7  Then  S  of . R . S  R .  image o f  »  7  By (ix)  a  l e f c  €  (  K  .  )  g  <  2  (»  g ( K )  ) ( r ) < (» (K))g(2)  (ii),  (»  g ( K )  )(r) -  we  conclude that  (»  g ( K )  s  o  u  g  , then  0  b  S  n  g  -(» ( ))(r) = ( ( *g(K)  l  s  (^ (K)^ ^ r  a  i  s  r  a  a  r  d  a  i  d  g ( K )  y  2  ' ' 1  2  )  •  c  a  ^  again •  r  g  a  (  )g(2))( )  2.1.3(ii)  K  c  g  r  g ( K )  ( ) ) ( ))( )  i  )  3  g  M  K  "  n  (K) •  3  i  o  R e (J»  W  s  n  ,  )(r)(r) < ( ( »  ( ^ ( K ) ^ g ( 2 ) - *g(K) g  a  image o f 2-  g  Since T ^  e  be a n y 2 - s u b -  7  Therefore  g (  (iv),  b  R e (!Hg^)(r)  ( i v ) , (* K)-) ( )  By  S  '  i s a homomorphic  7  Since  g  Let  (viii)  and  R  g  so  g(K)  .  g ( 2 )  R e (H (K))( )  homomorphic  r i n g of  -  2  class, i f c l a s s by  l  g  2.1.4. By so when and  r <_  2.2.2  radical  (viii)  we know t h a t  ( ^ ( ) ) ( ) ^- *g(K) r  g  K  arguments e x a c t l y p a r a l l e l i n g  c a n be u s e d  t o show t h a t  (^ (K)^^^  2.2.1 : i s  a  g  class.  Q.E.D.  I n P r o p o s i t i o n 2.4-3 G  i s a semi-simple K-class  class.  Unfortunately  generalized  K-classes  b e l o w we 1^  then  s h a l l prove that i f i s a generalized  the converse i s f a l s e ^g(K)  ^  o  r  w  h  i  c  :  h  ^he  since  K-  there are  class of  **g(K)  - 43 -  semi-simple and  34  group for  x  R  i s torsion free  +  x e R  x = 0).  e Q  n).  s  e  i f and o n l y i f h  Q = t h e r i n g o f r a t i o n a l numbers.  m  i "  s  such that  Therefore,  I  of  <x>  nR = (0)  fora l l  i  m  P l e  rings  c a n be  If homomor-  f o r some n o n - z e r o  x e Q , <x> Q e ** (2)  is  '  g  ^g(2)  ^o t h e c l a s s o f  2-class.  i s not a  PROPOSITION: K >_ 2  c l a s s and  be a c a r d i n a l number.  G = G(K)  ft = t h e c l a s s o f a l l r i n g s class  R e 34  the a d d i t i v e  f o r some p o s i t i v e i n t e g e r  On t h e o t h e r hand  Let simple  Let  i s , a ring. R  semi-simple.  2.4.3  hx = 0  (that i s ,  such that  mapped t o a n o n - z e r o r i n g o f f i n i t e c h a r a c t e r i s t i c  integer  ^g(2)  i f  let K = 2  F o r example, R  then every non-zero i d e a l  phically (that  i s not a K-class.  be t h e c l a s s o f a l l r i n g s  a l l  then  rings  of a l l rings  the  class  34  (0)  4 T e 34  R  T  =  a  a  (K)( )  the f o l l o w i n g  i s a K-ring  G .  tt ( )(T) g  n  e  r  e  Moreover,  "If 4  K  w  34 = t h e  ^ (°) •  condition:  then  i s a semi-  ( K ) ~ g(K)  R  g  G  W  g  which are not i n  such that  satisfies and  then  If  (0)."  Proof: We  shall  b e g i n by s h o w i n g t h a t R e "J^.  Suppose t h a t homomorphic image o f K-subring  T 4 (0) o f  R  . R'  Then  -  Let  XIQ R'  =  g  (K) '  be a n o n - z e r o  R' <| G = G(K)  such that  R  T $ G .  so there Therefore  isa  - 44 -  R  €  R  g(K) ' R e R r\ g(K)  Suppose m o r p h i c image o f R'  Therefore, e U  R  .  Then t h e r e  Then U  which i s i n = g(K) > R  £  i  so  s  T  of  R' <£ G .  R  is in  G , so  .  ^g(K)  Let  Then t h e r e is, a  ^g(K) '  =  R'  be a n o n - z e r o homo-  i s a K-subring  i s n o t 1/^s.s.  T  g(K) ' *  n  o  t  s  (  K  ) = I A Q .>  s  S  o  of  R  7  Since  e 34 .  T  T  Therefore,  *g(K) ' Since  34 / . s < R . g(K) g(K) T  To condition, ring. R  R  R , that T  that  R e ^-g(K) *  m o r p h i c image o f  R  i s a K-subring  = ^g(K) *  a  Suppose  G  be a n o n - z e r o homo-  T $ G = G(K)  Then  Now we s h a l l p r o v e  1X  L e t . R--  no n o n - z e r o homomorphic image o f  •  G  R  T e R .  such that  • .  Tr  'g(K)  M  g  Therefore  see t h a t  R  T e 34  34 = R • g(K) g(K) t  v  S  34  satisfies  the d e s i r e d  (0) / T e 34 a n d t h a t g  ^ ( T )  g(K) ' *g(K)( ) ^ T  Thus b y 1 . 4 . 2 ( i i ) ,  <_ ft .  the class  suppose t h a t  Since =  a  ^ (0) . *  T  h  i  s  c  o  T  i s a K-  So, b e c a u s e m  P  l  e  t  e  s  t  h  e  proof. Q.E.D.  The  c o n d i t i o n o f the proceeding p r o p o s i t i o n  be  u s e f u l i n some o f t h e f o l l o w i n g r e s u l t s .  it  easy to r e f e r  to this  closely associated  condition  I n order  (and t o a n o t h e r  w i t h i t ) we make t h e f o l l o w i n g  will t o make  condition definition.  - 4  2.4.4  -  DEFINITION; For  34  5  may  each  satisfy  K > 2  c a r d i n a l number  . a class of rings  e i t h e r of the f o l l o w i n g :  Condition  r(K) :  If  (0)  s(K) :  If  and  g  T  ideals  i s a K-ring of  T  then  it  and a l l non-zero  c a n be  2.4.5  that  T  in  the s t a t u s  *g(K) (  So we make t h e f o l l o w i n g  i s a non-zero I e * ( ) g  a l l classes We  K = 2  in  34 ,  a n d  2.4.3,  of i d e a l s  ^  i d e a l s °f  defintion.  K  i s absorbent  K-subring of  then  of rings  K  ( l )  R  e  K =  which  • are  g  ( -j K  i f and o n l y i f and  (0) / I«3 T  , f o r a l l rings  R  34 .  are mainly i n t e r e s t e d  and when  M  R  <_  i n t h e s i t u a t i o n when  In Chapter  I V we  all  cardinals  ing  q u e s t i o n i s whether o r n o t a l l " c a r d i n a l  bent.  rings  DEFINITION:  whenever  and  about  a r e g e n e r a t e d by K - s u b r i n g s  A c a r d i n a l number  such  homomorphically  order to prove a converse of P r o p o s i t i o n  such s u b r i n g s ) .  K-ring  T e 34 .  i s n e c e s s a r y t o know s o m e t h i n g  which  is a  .  K  mapped o n t o n o n - z e r o  In  T  » ( ) ( T ) 4 (0)  then Condition  T e 34  s h a l l prove  are absorbent.  that  An i n t e r e s t -  numbers  are absor-  -  2.4.6  PROPOSITION: If  any  46 -  class  ^ g(K)  s e m i  K  i s an absorbent  of rings ~ - - P s  i  m  then  cardinal  G <_ G(K)  >_ 2  where  and  d  is  G = the c l a s s of  rings.  l e  Proof: Assume If  T £ G  bent  R e G . (0) 4  then  e ^g(K)  contradiction/  so  s  Let  T  / (K)( ) T  =  be a K - s u b r i n g o f 1  g  o  R  i  T e G .  s  n  o  '  S  i  n  c  e  ^g(K) * '  t  s  Hence  s  K  T  l  h  i  absor-  s  s  R .  i  s  a  R e G(K) . . Q.E.D.  2.4.7 THEOREM: Let G  K  be a n a b s o r b e n t  i s a semi-simple  class,  cardinal  number  >_ 2 . I f  then:  I  G = the class semi-simple class  rings  of rings  Condition  of  Jt  *g(K)' f o r some satisfying  r(K).  Proof: Let let  G  K  be a s e m i - s i m p l e Assume  the  be an a b s o r b e n t  class  of  that  G  cardinal  number  _> 2  and  T h e n b y 2.4.3  G =  class. i s a K-class.  M^^^s.s. rings  where  H  satisfies  Condition r(K)  - 47 -  Conversely, assume r i n g s where  tt  G = the c l a s s of  s a t i s f i e s Condtion  r(K) .  G <_ G(K) . We need only show that and  (0) ^ I <Q R .  let  subring  T  Condition  of  I  r(K)  If I e  such that ,  M g  f  v  S  K  h  e  there i s a K-  n  s  o  Since T  g  c o n t r a d i c t i o n s i n c e we assumed that I k tt so * g( ) •  t  K  T e Ji .  T  Then by 2 . 4 . 6  G(K) <_ G . L e t R e G(K)  ( )  * (K)( ) ^  tt^^s.s.  £  tt  •  G  T  R e G(K) .  h  i  satisfies s  i  s  a  Therefore  R e G Q.E.D.  2.4.8  COROLLARY: Let  K  I f <J  be an absorbent c a r d i n a l .  rings s a t i s f y i n g  condition  (A) then the c l a s s of  i s a class / t r  xS.s.  rings i s a K-class.  Proof: The c l a s s  p$ s a t i s f i e s Condition  satisfies  (A) , i f T e J  since  $  then  T e d(j g  K  and  r(K) . T  In f a c t ,  i s a K-ring  • Q.E.D.  For absorbent c a r d i n a l numbers  K , Theorem 2 . 4 . 7  answers the q u e s t i o n , "When i s a semi-simple c l a s s a K - c l a s s ? " The next theorem answers the q u e s t i o n , "When i s a K - c l a s s a semi-simple  class?"  - 48  2.4.9  -  THEOREM: Let  a K-class  K  be an absorbent c a r d i n a l  >. 2 .  If  ft  is  then:  ft i s a semi-simple c l a s s i f and Condition  s(K)  only i f  ft  satisfies  .  Proof: Let  K  be an absorbent c a r d i n a l  >. 2  and  ft  a  K-class. Assume fies  ft  i s a semi-simple c l a s s .  c o n d i t i o n (F) so c e r t a i n l y  ft  Conversely, assume t h a t s(K)  .  tainly fies  Since ft = ft(K) , ft  satisfies  R  r i n g i n ft . R  If  R $ ft =  such t h a t  T £ ft .  homomorphic image of  K  satis-  Condition  satisfies  s(K).  Condition  i s s t r o n g l y h e r e d i t a r y so  R  To  show t h a t  i s a r i n g and  I  ft(K)  then there  Since I  of  ft  cer-  satis-  every non-zero  ft  i s a K-subring  satisfies  Condition  T  such t h a t no  i s i n ft .  Because ft =  t h a t every non-zero homomorphic image of  a K - s u b r i n g which i s not i n ft . ft = the  ft  can be homomorphically mapped onto a non-zero  there i s a non-zero i d e a l  implies  ft  condition (E).  c o n d i t i o n ( F ) , suppose  i d e a l of  of  ft  satisfies  Then  Thus  I  c l a s s of a l l r i n g s which are not  i s absorbent,  (I)T-, e ft / ^ v . 'R g(K)  But  s(K),  non-zero ft(K) I  this  contains  i s ft / -, • g(K) Tr  i n ft .  T  Now,  this contradicts  where since our  V  assumption t h a t every non-zero i d e a l of  R  can be homomorphi-  cally  mapped o n t o Thus,  ReB.tt  is  a non-zero  ring  in  satisfies  both  condition  tt  a semi-simple  tt(K)  = tt .  Therefore  ( E ) and  (F),  so  class. Q.E.D.  Usually, class  tt  hence  tt  plies  that  the  vided  that  K  s  2.4.10  -  g(K) r  x  r  definitions classes  , the  s  g(K) fv  »  g  K  v N  class  N  /  of  considering  satisfies  .  ^g(K) " '  an absorbent be u s e f u l  for  which  in  condition  s  r  i  n  SS  is  cardinal).  The  )  g ( K )  =  g ( K )  K-  and  this  a K-class  investigating  (tt  (A)  2 . 4 . 8 we s a w t h a t  In s  a generalized  im(pro-  following  generalized  tt  K-  .  g(K)  DEFINITION: For  satisfy  tt  (tt/ ) /„ g(K) g(K)  is  )  class  v  will (  w h e n we a r e  either  Condition  each c a r d i n a l of  the  r(K)  number  s(K)  :  If  (O)  e  tt  :  ^ T  there  is  a class  and  of  of  T  such  If  T  is  a K-ring  cally tt ,  r(K)  of  that  T  mapped o n t o  then  and  T  T  rings  is  a non-zero  L  K-subrings  Conditions  ,  may  following:  then  Condition  K  K-ring  K-subring  L e t t and a l l  c a n be  a  f v  *  .  non-zero  homomorphi-  non-zero  rings  in  e tt .  s(K)  seem t o be  slightly  - 50 -  stronger fact,  than Conditions  if  implies  K  Is  an a b s o r b e n t  Condition r(K)  that  in this  L e f t  /rr-\ g(K)  r(K)  case I f  then the  . L  and  ideal  this  n o t so c l e a r .  (L)  that  if  satisfies  K  However,  one n e e d o n l y  R D  r(K)  notice R  and  T  C o n d i t i o n s(K)  c a r d i n a l and  If  In  e ft /,N . g(K)  i n 2 . 4 . 1 2 we a r e  i s an a b s o r b e n t Condition s(K)  Condition  a K - s u b r i n g of a r i n g  The r e l a t i o n s h i p b e t w e e n is  respectively.  c a r d i n a l then  To see is  s(K)  ft.  ft  satisfies  C o r r e s p o n d i n g t o 2 . 4 . 7 we  able <_  and to  s(K)  prove  ft(K)  then  C o n d i t i o n i~(K)  ft .  prove:  2 . 4 . 1 1 THEOREM: Let G  is  K  be a n a b s o r b e n t  a semi-simple  class,  c a r d i n a l number  class  ft / „ \ S . s . is a K-class  'G g ( K )  and  if  If  then:  G = the  G  >_ 2 .  and o n l y  G  if  class  F(K)  rings for  of r i n g s  satisfying  of some  ft  Condition  .  Proof: Let  K  be a n a b s o r b e n t  cardinal  > 2  and l e t  G  - 51 -  be a s e m i - s i m p l e  class.  Assume Then by 2 . 4 . 3 ,  G H  i s a K-class =  G  R g  (K)  r i n g s which a r e not i n by 2 . 4 . 2 ( v i ) , satisfying 3  g  (K)  h  e  G .  ^g(K)  =  condition  satisfies  r(K)  R  w  r  e  or  s  t  Z.  (A),  class of a l l  e  f  o  m  i  v  r  class of rings  e  (A). Therefore,  condition  h  (R / ^ s ) s •= R / \ g(K)'g(K) g(K)  1/t  so  3"  = ^g(jc)  and, s i n c e  satisfies  Condition  Q  certainly  . C o n v e r s e l y , assume t h a t  rings  and t h a t  show t h a t Suppose  G  14  satisfies  R { G .  that  satisfies  (and  hence  M  g(K)( )  W  there L e that  (K) •  w g  G  =  M g  (K) '  s  o  b  y  2 , J +  *  2  (  i v  )  Thus =  sufficient  s  shall  G <f G(K) . isa  L  L  ( I L G ^ K ) V-Q l  we  T e 34 .  Thus,  G(K) <_ G .  l 1 z  N  so t h e r e  i s a K-subring  N e x t we must show t h a t  *\JL  2.4.6  ^ (°)  34' / , , s . s . g(K)  First  such that  R  R) s u c h t h a t  So we c o n c l u d e  (K)( ) R  g  Condition' r(K)  of  R £ G(K) .  of  r(K) .  By P r o p o s i t i o n  Then  T  G = the c l a s s o f  Condition  i s a K-class.  non-zero K-subring 34  =  R  Now, **  d4.Q)g(K) ~ M»G '  and  Since  of G  T  so  G = G(K) . *  N  O  W  t o show  that  *g(K) ^ ^ g ( K ) ) g ( K ) * Suppose n o n - z e r o homomorphic image K-subring  T'  of  R'  of  R e R  such that  .  S g  (K)  a  n  d  l  e  Then t h e r e  T' e 34 .  t  R  b  e  a  i s a non-zero  Since  34  satis-  f i e s C o n d i t i o n r ( K ) , T' c o n t a i n s a n o n - z e r o K - s u b r i n g .s L' such t h a t L ' e 34 . . T h e r e f o r e R e (34 _. ) . ' Thi£ g(K) g(K) g ( K ) f v  /t  ;  - 52 -  completes  the proof.  Q.E.D.  Before proving Theorem 2 . 4 . 9 , Conditions  2.4.12  we n o t i c e  s ( K ) and  a theorem  which  corresponds to  t h e f o l l o w i n g r e l a t i o n s h i p between  s~(K) .  PROPOSITION: Let  K  be a n a b s o r b e n t  be a c l a s s o f r i n g s Condition  s(K)  s u c h t h a t ft <_ ft(K) .  then  ft  >_ 2 , a n d l e t ft  cardinal  satisfies  If  Condition  ft  satisfies  s(K) .  Proof: Let  K  be a n a b s o r b e n t  be a c l a s s o f r i n g s  ft  satisfies  ft = t h e c l a s s o f a l l r i n g s w h i c h  T \ ft , t h e n by C o n d i t i o n  and  non-zero  K-subring  phically  mapped o n t o a r i n g i n ft .  L  i s absorbent,  of. T  (L)  e  homomorphically  ft <_ ft(K) , t h i s Condition  s(K) .  Thus  ft  .  L  If  T  is  i sa  c a n n o t be homomorL e g(£) • R  Thus  g(K)  mapped o n t o a n o n - z e r o  establishes  s(K) . L e t  s(K) , there  such that  T be  Condition  a r e n o t i n ft .  a K-ring  K  a n d l e t ft  s u c h t h a t ft <_ ft(K) .  Assume t h a t  since  >_ 2  cardinal  (L) v  ft(K)  the r e s u l t - t h a t  ft  m  Now,  cannot  'T ring. must  Since satisfy  Q.E.D.  - 53 -  Notice class  of A  class  satisfies  K  t h a t , by 2.4.6, t h i s  / ,\S.s. rings g(K)  satisfies  T  Condition  s(K)  .  implies  that  Condition  s(K)  (Provided,  i f the then  this  of course,  that  i s absorbent).  Now, c o r r e s p o n d i n g t o 2.4.9 we p r o v e :  2.4.13 THEOREM: Let a K-class G  K  be a n a b s o r b e n t  (V ) ( ) G  >_ 2 .  If  G is  then:  i s a semi-simple  and  cardinal  g  K  =  class' I i f and o n l y  1A  J  G  G  satisfies _ I Condition s(K)  if-  Proof: Let be  a  K  be a n a b s o r b e n t  G  )  G  ( ) K  = UQ •  G  ft  Suppose t h a t  T  so,  satisfies  since  K-subring  ft L  and l e t  G  of  w g  to non-zero r i n g s s"(K)  .  in  and  Condition  (K)^) of  r(K)  ft ^s.s. g  Condition  T k G .  such that T  c l a s s and t h a t  G = the c l a s s o f  which s a t i s f i e s  i s a K-ring  non-zero K-subrings  Condition  i s a semi-simple  By 2.4.11,  r i n g s f o r some c l a s s  all  >_ 2  K-class. Assume t h a t  (U  cardinal  Then  , there  r(K)  .  * (K)( )  ^ (°)  T  g  i s a non-zero  L e ^g(^)  *  T  n  u  s  not  c a n be h o m o m o r p h i c a l l y mapped  G(K) = G .  Therefore,  G  satisfies  -  Conversely, T h e n , by G  2.4.12,  G  %  Now,  tt  and  g(K)  the p r o o f  Suppose  R  of  R  Now,  R'  R'  such  that  T  there phic R  6  ( 3  class.  Now,  e *g(K)  a  n  d  R  .  i s a K-subring image, o f  L  satisfies  tt  2.4.7,  '  i  L  . so  which s a t i s f i e s (*  G  .  there  G  of . T  is in  the  g  (  K  )  )  g  (  K  r(K)  < *  )  ^g(K)  —  class v  g  e  To  ^g(K)^g(K) image T  of  C o n d i t i o n s~(K),  s u c h t h a t no L  •  )  i s a K-subring  satisfies  Thus  K  of  .  ;  (  s(K)  2.4-9,  , so by  n o n - z e r o homomorphic  s a  Since  Condition  G =  n e e d o n l y show t h a t  ^ G = G(K) $ G  by  2.4.2(iv),  by  we  G  C o n d i t i o n s(K)  &  complete  .  assume t h a t  r i n g s f o r some c l a s s =  a  4  satisfies  i s a semi-simple  tt / „ v S . s . g(K)  5  n o n - z e r o homomor-  tt g(K)  .  Therefore,  g(K) g(K) * }  Q.E.D.  2.4.14 COROLLARY: Let be  any  class  satisfies a  g(K)  and  s , s  '  tt(K)  K  be  an  of r i n g s .  absorbent There i s a  c o n d i t i o n (A) and r  i  n  S  s  l  f  a  n  d  cardinal class  such that  only i f  tt  tt  >_ 2  , and  of r i n g s =  satisfies  the  ft  class  let  tt  which of  Condition  s"(K)  = tt .  Proof: Since  any  (A) a l s o s a t i s f i e s immediately  class  of r i n g s which s a t i s f i e s  Condition r(K)  f r o m 2.4.11 and  , the  corollary  condition follows  2.4.13.  Q.E.D.  55  We c o n c l u d e those g e n e r a l i z e d cal  this  -  chapter with a r e s u l t  K-classes  *g(K)  f  o  r  w  h  i  c  h  concerning l  s  radi-  a  class.  2 . 4 . 1 5 THEOREM: Let a radical  K  be a n a b s o r b e n t  >_ 2  cardinal  .  I f ft i s  c l a s s and e i t h e r :  ( i ) ft < ft(K) o r (ii)  i f Reft  then a r i n g where  M  R  then  is  « ( g  (K) * *  w  s  g  s  i  ( R ) 4 (0)  K )  f  a  n  d  o  n  y  l  i  f  R  €  #|( ) K  = t h e c l a s s o f ft- s . s . r i n g s .  Proof: Let be  a radical (a)  K  be a n a b s o r b e n t  class.  Let  cardinal  > 2  and l e t  o£ = t h e c l a s s o f ft s . s . r i n g s ,  Suppose t h a t ft <.ft(K). Assume t h a t R R I  £ <?/(K)  then there  K-subring  So  /v\ g(K)  This  contradicts  and s i n c e  ring.  i s a non-zero K-subring  I (and hence,  L e f t .  L e f t  R e 5?(K)  i s a ft /„^s.s.g(K)  I = ft(T) 4 ( 0 )  such that e ft(K) .  R  Since K  ft  .  If  T  of  S i n c e ft < ft(K) , T) c o n t a i n s  satisfies  i s absorbent, '  our assumption,  a non-zero  condition (A), (L)„ e 'R  ft . g(K)  s o we must have  that  . C o n v e r s e l y , assume t h a t  R e ^ (K)  .  Then  -56 no K - s u b r i n g o f Suppose  that  R  is in  whenever  there  there  .  is  J u s t as i n ( a ) , t h i s Re  L  .ft ^s.s. g  of  R  I  If  such  £ J  (0)  K  such that  contradicts  R  (K)  that  « ( )(l) J g  (°)  =  '  R  .  So  L e f t .  our assumption.  QJ(K) .  C o n v e r s e l y , assume t h a t no K - s u b r i n g o f  R  g  (ii), of  **g(K)( )  ^ (K)^ ) ^  T  T h e n by  i s a K-subring  Therefore,  R  i s a K-subring  ft(T) = 1 / ( 0 )  so c l e a r l y  R e f t ,  Assume t h a t then  ft  R  is in  ft  R e ^f(K)  so c l e a r l y  M  .  Then  ( )( ) R  g  K  =  Q.E.D.  (°)  - 57  -  CHAPTER I I I ELEMENTARY RADICAL CLASSES  3.1  THE  ELEMENTARY RADICAL CLASSES We  classes  3.1.1  s h a l l b e g i n our  with a discussion  &  .&  . AND  r  FC.  study of elementary S  of  and  &-  radical  .  R  DEFINITION:  (i)  Sr x  i s the e R  a^, (ii)  Jo-  c l a s s of a l l r i n g s  , a^x  ...,  +  N  a^  e R  , x  If then c l e a r l y B <J A  a  K - 1  K-i>  R  € h  R'  eh  such that  each  f o r some - i n t e g e r s  a^'s  are R  0)  N  ,•  w h i c h depend  such that f o r  each  K—1  + a  integers  = 0  such that f o r  c l a s s of a l l r i n g s  K  x  + a^x  (not a l l the  i s the  R  ...  R  x  +  •••> ] _ and .  Bet?  . .. + a-^x  = 0  f o r some  a  w h i c h depend on  x  .  R'  i s a homomorphic image o f  Suppose t h a t  A  is a ring  e tr'  .  Let  and  A/B  x  e A  R  and .  Then  N a^x for  +  some i n t e g e r s  Since zero,  B  e ^  such  ...  a^,  , there  that,  + a.jX  = b  . .., are  a^  e B (not a l l of which are  integers  c^,  • • •,  zero).  , not a l l  on x,  - 58 -  thus  £ j=l  implies Sr  c,( E 1=1  a. • x ) " = 1  J  3  that  A e ^-  .  E £ H = l j+i=H  c .a.x J  11  = 0 .  This  1  By t h e d e f i n i t i o n  i t Is clear  that  Ss- i s a r a d i c a l c l a s s .  i s a n e l e m e n t a r y c l a s s s o b y 2.3«'l  A s i m i l a r argument shows t h a t  &  Both  elementary c l a s s e s which  ir a n d  &  R  are clearly  i s also a radical class.  R  tain  the class of a l l n i l p o t e n t rings  sees  that  j8 < 71 <_  <_ ^  ).  (in fact,  Combining  con-  one e a s i l y  t h e above  remarks  w i t h 2.3-4 we h a v e :  3.1.2 PROPOSITION: Jr  3-1.3  and  js- a r e s p e c i a l elementary r a d i c a l c l a s s e s . R  LEMMA: If  I <J <?[X]  and  I /  (0) then  $[X]/I  e tr  .  Proof: (0) 4 K  Let  &[X) - and- f'(X) e  g(X)  / 0 , g(X) e I .  L e t 0 ^ , .. ., a  g(X)  .  are a l l algebraic  Since  = f(o.^)  .  the  Thus t h e r e  K  @[X]  .  Choose  be t h e r o o t s o f numbers, s o a r e  a r e non-zero polynomials  &[X]  h^(X) e  K such that  h.(y,) = 0 .  L e t h(X) = J J h.(X) , t h e n • i= 1 . •  are  integers  Consider roots  a^,  a  h ( f ( X ) ) = l(X)  a,,  there  1  1  C L . of  such t h a t  M  e  Q[X]  .  h(X) = a^X + ... + a^X ^. 1  Now  g(X) , s o t h e r e  l(a ) ±  =0  for a l l  i s a polynomial  d(X)  - 59 -  w i t h r a t i o n a l c o e f f i c i e n t s such t h a t  -t(X) = d ( X ) - g ( X ) .  S i n c e t h e c o e f f i c i e n t s o f d ( X ) a r e r a t i o n a l t h e r e i s an integer Now  n  such t h a t  n«d(X)  has i n t e g e r  n - h ( f ( X ) ) = na _f(X) + ... + n a ( f ( X ) ) ]  Thus,  na F(xy +  fore,  fax]/I  M  M  + na M (f7xT) M  1  €U  coefficients.  =  0  in  = nd(X).g(X) e I . f?[X]/I .  There-  • Q.E.D.  The elementary r e s u l t s about a l g e b r a i c  numbers  used i n 3 . 1 - 3 can be found i n Chapter 9 o f N i v e n and Zuckerman Lemma 3'1«3 i m p l i e s  [12].  r a d i c a l class  that  &> i s t h e l a r g e s t  elementary  (except f o r the c l a s s o f a l l r i n g s ) .  I n the  following proposition  we c o l l e c t some i n f o r m a t i o n about t h e  r e l a t i o n s h i p between  <$3-  classes  3.1.4 (i)  jfir  R  and t h e well-known  radical  l i s t e d i n Chapter I .  PROPOSITION: I f •.tt'i s an e l e m e n t a r y r a d i c a l c l a s s and tt' i s n o t the  (iii)  ,  c l a s s o f a l l r i n g s then tt' <_ $r .  Neither  nor  Assume t h a t  tt'  i s related to  )fl  or  J .  Proof: (i)  i s an e l e m e n t a r y c l a s s and tt' i s  not t h e c l a s s o f a l l r i n g s . if  fax]  Then  fax]  tt'  ; for  e tt' then e v e r y r i n g g e n e r a t e d by one  - 60 -  element ( b e i n g a homomorphic image of M  , so  7  If  U'  x e R  i s the c l a s s o f a l l r i n g s . then  <x> ~  6>[X] £ «' , I / (0) fore (ii)  If  R efr R  W <_ &  , so  Yl <L & R <_ $r • and  ^£  .  & (iii)  £  R  x e R  then  <  7  There-  %  .  f?[X]/(2X)  = 0 for  x^^ n  Certainly  x  then,  f i e l d s are i n  Hence )t £ «(r £ &  & i  ft  ,  g  J  R  ±  tfg  , &  F o r example, c o n s i d e r any f i n i t e F e  but  R To see t h a t  and  R  b u t i s i n ir  R  F i  i J  and j £  field  F .  so  field  R  £ • J  .  F e &-  Yl  and F i J . g c o n s i d e r the r i n g r  J £  f o r m a l power s e r i e s i n one i n d e t e r m i n a t e  S 1=0  M.  Since  <x> e U  However, a l l f i n i t e  The r i n g  R e  .'  n(x) .  &r b u t n o t i n  Let  & .  Clearly  and'  .  so by 3 . 1 - 3  some non-zero i n t e g e r  R  (?[X]/I e  i s a n i l r i n g and  d£-  (?[X]) i s i n  x  R  of a l l  over the r a t i o n a l  Q . Let R be the i d e a l o f R c o n s i s t i n g o f a l l i a.x e R such t h a t a = 0 . I t i s well-known (and easy 7  1  0  to p r o v e ) t h a t  R  e J .  7  However, i f a^,  a^  are i n t e -  K  gers,  a^x + ... + a^.x  Thus,  R  7  j£  Let  .  unless  a^ = ... = a^. = 0 .  T h i s example shows t h a t  Q(x)  i n an i n d e t e r m i n a t e  / 0  J £ Jr  R  and  be the f i e l d of a l l r a t i o n a l f u n c t i o n s x  over the r a t i o n a l f i e l d .  Q .  Then  - 61  R = (Q(x))  2  -ft  .  a^,  . .., a^-  $ the r i n g of  However,  a  then  2x2  R  m a t r i c e s over  (x) , i s i n  R £ $ r , because i f  and  a r e i n t e g e r s such t h a t  fx  OK  iVo  oJ  5  -  ^  •'•  +  +  /x  0  K\o  0\K  oJ  a  a^x + ... + a^x^" = 0  in  fO  0\  - VO Q(x)  o7 so  a^ = ... = a ^ = 0 Q.E.D.  We now radical class  3.1.5  t u r n t o a b r i e f c o n s i d e r a t i o n of the  PC .  DEFINITION: . (i)  Let  of a l l r i n g s  R  tive integer  a(x)  (ii) f o r each  p  be a prime number.  such t h a t f o r each  FC  x e R  such t h a t  FC^  x e R  I s the c l a s s  there i s a p o s i -  p ^ ).x = 0 . a  x  i s the c l a s s of a l l r i n g s  R  t h e r e i s a non-zero i n t e g e r  such t h a t  n(x)  such t h a t  n(x)«x = 0 . Let d i t i o n (A). are i n a .  PC  p  p  be a p r i m e .  Suppose t h a t .  But s i n c e  Let  B^  Clearly A  x e A , then  p x e B e PC  PC^  and both p°x  satisfies B  e B  , P (p x) = 0  and  con-  A/B  f o r some i n t e g e r f o r some i n t e g e r  sr  jS .  Therefore  p^^x  = 0  so  A e PC  .  Now,  i t i s clear  3?  from the d e f i n i t i o n t h a t 2.3*1,  FC  PC^  i s an.elementary c l a s s so by  i s an e l e m e n t a r y r a d i c a l c l a s s .  A similar  - 62 -  argument shows t h a t  PC  i s an e l e m e n t a r y r a d i c a l c l a s s .  '3.I.6 PROPOSITION: (i)  FC  (ii)  i s an e l e m e n t a r y r a d i c a l c l a s s ,  F o r each prime  p , FC  i s an e l e m e n t a r y r a d i c a l  class. (iii)  R , PC(R) = Q (PC ( R ) : p  For a l l rings  i s a prime}.  Proof: We have a l r e a d y seen t h a t t h e c l a s s e s mentioned i n ( i ) and ( i i ) a r e e l e m e n t a r y r a d i c a l c l a s s e s . To see t h a t ( i i i ) i s t r u e , n o t i c e t h a t  R e PC i f  4.  and o n l y i f t h e a d d i t i v e group i f and o n l y I f R  +  R  i s a p-group.  i s t o r s i o n , and R e FC P a r t ( i i i ) o f the p r o p o -  s i t i o n now f o l l o w s f r o m a well-known r e s u l t about t o r s i o n groups. Q.E.D. 3-2 THE ELEMENTARY RADICAL CLASSES J" R e c a l l that  i s the lower r a d i c a l c l a s s d e t e r -  mined by a l l n i l p o t e n t r i n g s some r i n g tains  *J  R  AND £>' .  N  such t h a t  w i t h D.C.C. on l e f t i d e a l s .  N = 77(R) f o r T h i s c l a s s con-  which i s t h e lower r a d i c a l c l a s s determined by t h e  c l a s s o f a l l zero simple  rings.  I n t h i s s e c t i o n we s h a l l c o n s i d e r t h e c l a s s e s and fa ' . We b e g i n w i t h the f o l l o w i n g r e s u l t about t h e  rj'  -  class  <$  -  63  .  3 - 2 . 1 PROPOSITION: J  = js n P C  .  Proof: L e t ft = the c l a s s of zero s i m p l e r i n g s . then the a d d i t i v e group S  S  T h e r e f o r e , ft <_ PC  and so  r a d i c a l c l a s s determined Suppose t h a t R  which i s i n  /3 D PC  Suppose R e PC  J <_ PC  jS fl PC £ J  .  but i s I  of  f o r some prime  because Since  , i ' e PC  <j 1  .  J  J  J  9  I s the  <. P ,  j  semi-simple. R  such t h a t 7  , and  such t h a t  1.1.7)  Therefore  I .  Since  and hence j3 fl FC <. J  px' = 0 .  R e )3  = ( 0 ).  I  x' / 0  <x'>  lower  < jS fl F C .  Since  p  but  p .  Then t h e r e i s a r i n g  Thus, t h e r e i s a prime  Since  and an element (l') =( 0 ) , 2  i s i s o m o r p h i c t o the z e r o r i n g on the  c y c l i c group of p-elements. (see  elements  i ' i s a non-zero homomorphic image of  x' e I ' <x'>  p  by ft .  t h e r e i s a non-zero i d e a l  S e ft  i s a s i m p l e a b e l i a n group; so,  i s the c y c l i c group of  +  If  I e  But then  <x'>  e ft so  I e ft^  «jf . T h i s i s a c o n t r a d i c t i o n .  , so  = j8 fl FC . Q.E.D.  3.2.2  THEOREM:  D ' = $ ' = Ti n PC  - 64 -  Proof: J'  F i r s t we s h a l l show t h a t  =Jl  j8' < yi'  that  f o r i f <x> e Yl  then  Since, J  j'  <  <x>  ; and t h a t  i s n i l p o t e n t so  and  <_ FC' = FC  Re  x  K  = 0  and Nx = 0 . Thus  <x> e Yl , < > e d  since  Yl n FC <_ Since <J '  <x> ^¥1'  = H '  J  <  £)  ,  0  <  x  >  i  F  and N  C  such  has D.C.C. and s o ,  Suppose  < / ' < £ ) ' .  . The r i n g s  <x>  K  <x>  [7]).  <x> •  L  <x> e  B'  must be n i l s i n c e e  t  <  x  = <x>/FC<x> .  >  <x>/<x>  2  First  ^ FC . Suppose  — 2  nx e <x> _  so  j p ' = Yl D FC .  so  we s h a l l prove t h a t t h e r i n g  —  Thus  and J ° < FC .  (see Lemma 28 i n D i v i n s k y  x  Therefore,  B'  < P>' ,  Yl D FC , and l e t x e R . Then  i s f i n i t e since there are p o s i t i v e integers  but  yi'  Yir\ FC .  Suppose  that  Notice  <x> e j3 .  J < 0  = j3 n FC (by 3.2.1),  g / ' = £' = "ft  Therefore,  P <Jl  since  = Yl n FC .  . Then t h e r e a r e i n t e g e r s  a^, . .., a ^  such t h a t  M  nx =  S a. x . By s u c c e s s i v e s u b s t i t u t i o n s f o r nx we see 1=2 t h a t n^x e <x>^'"'" f o r a l l p o s i t i v e i n t e g e r s K . Since 1  +  j£  <x>  i snilpotent,  n x = 0  f o r some i n t e g e r  <x>  i s a r i n g of c h a r a c t e r i s t i c 0  T h i s i s a c o n t r a d i c t i o n on s i n c e  K , and s i n c e  t h i s implies that  x = 0.  <x> ^ FC . T h e r e f o r e  2 ' <x>/<x>  FC  so  <x>  can be homomorphically mapped t o a  zero r i n g o f c h a r a c t e r i s t i c z e r o .  Since  <x>  I s generated  - 65 -  by one element t h i s r i n g must be i s o m o r p h i c t o C z e r o r i n g on the i n f i n i t e c y c l i c group. C" e ©  . B u t c" <j: §b  Since  the  <x> e $  ,  (see Theorem 14, D i v i n s k y [7]) so  this i sa contradiction.  Hence ^)'  <_ J*'  so  §b' = ^f'  arid t h e p r o o f i s complete. Q.E.D.  3.3 CLASSES tt FOR WHICH tt' =71 In  [9] Goldman d e f i n e s a H i l b e r t r i n g t o be a  commutative r i n g  R  w i t h i d e n t i t y such t h a t  f o r a l l homomorphic images R  .  R'  J ( R ' )  =  Yl(R')  o f R . He proves t h a t i f  i s a H i l b e r t r i n g t h e n so i s t h e p o l y n o m i a l r i n g  R[X] .  S i n c e the r i n g o f i n t e g e r s i s c l e a r l y a H i l b e r t r i n g the f o l l o w i n g  proposition i s a special  case o f Goldman's  theorem. •.  3.3.1 PROPOSITION: If J(R)  =  R  i s a r i n g g e n e r a t e d by one element t h e n  YKR) •  3 . 3 . 2 THEOREM: If  j3 <_tt< FF  t h e n tt' = 71 .  Proof: Suppose t h a t tt i s a c l a s s J3 <_ W <_ FF .  of rings  such t h a t  -  Since  <x>  -  Y\ = /3' <_ 34' .  j8 <_ 34 ,  Assume t h a t  F F ' ^_ "Yl .  such t h a t  >£(<x>) = (0)  e FF'  and so, s i n c e [7]  66  Then t h e r e i s a r i n g By 3-3-1  .  i s commutative, by Lemma 87 i n D i v i n s k y  <x>  F(<x>) = (0)  .  I t follows that  <x>  i s a subdirect  of f i e l d s each of which i s g e n e r a t e d by one element.  i s g e n e r a t e d as a r i n g by one element must be <y>  be a f i e l d and suppose  c h a r a c t e r i s t i c zero. tionals. so t h a t  Choose  Then  <y>  contains  1/p  let  = c^y + ... + c y n  a^y^ " = Ny - a-^y 4  1  tive integer  .  of m i n i m a l degree  a  ^/p  p  = b^y + ... + b y  Let  p  and  substitutions for  .  k  k  does not d i v i d e any of the  c o e f f i c i e n t s i n this expression  .  we see t h a t f o r some p o s i -  0 = Na£ - Na£ = ( a ^ - p N b ^ y + ... + ( a £ Since  <y>  a^, ..., a ^  By c o n t i n u e d  - ... - a^.^Y^  2  L ,  n  i s of  a copy o f the r a -  i s a non-zero i n t e g e r i n  be a prime which does not d i v i d e any of  which  finite.  <y>  f ( y ) = a-^y'+ ... + a^y^  N = f(y)  sum  To r e a c h  a c o n t r a d i c t i o n i t i s s u f f i c i e n t t o show t h a t any f i e l d  Let  (0)  J(<x>) =  are  a  i  0 .  +1  But then - pNb )y k  k  .  not a l l of the Since  <y>  has no  p r o p e r d i v i s o r s of z e r o i t f o l l o w s t h a t t h e r e a r e i n t e g e r s d-p and  d^  with  I <_ k .  d  4 0  1  But- then  such t h a t  d^ = -d^y - . .. - d^y  c o n t r a d i c t s the m i n i m a l i t y of Thus  <y>  d^y + . .. + d^y  = 0 which  k .  i s of f i n i t e c h a r a c t e r i s t i c and  since  -  <y>  67 -  must be a l g e b r a i c i t f o l l o w s t h a t  <y>  T h i s i s a c o n t r a d i c t i o n and so  i s finite.  FF'  <_yi  . Q.E.D.  T h i s theorem may be p a r a p h r a s e d i n the f o l l o w i n g way,  "A r i n g  R  i s n i l i f and o n l y i f no s u b r i n g o f R  which i s g e n e r a t e d by one element can be homomorphically mapped onto a f i n i t e  field".  3-4 ELEMENTARY RADICAL CLASSES WHICH ARE  = FC .  We w i l l b e g i n t h i s s e c t i o n w i t h a d i s c u s s i o n o f t h e e l e m e n t a r y r a d i c a l c l a s s fi' . T h i s r a d i c a l c l a s s i s u n r e l a . t e d t o a l l o f the well-known r a d i c a l c l a s s e s l i s t e d i n Chapter 1.  In fact, a l l  S'  rings are H  semi-simple.  This r a d i c a l  p l a y s a c e n t r a l role, i n o u r d i s c u s s i o n s c o n c e r n i n g r a d i c a l c l a s s e s which c o n t a i n o n l y T i s e m i - s i m p l e r i n g s . '  3.4.1  DEFINITION: £  a l l rings  i s the c l a s s o f a l l Idempotent r i n g s ( t h a t i s , R  such t h a t  R = R ). 2  Let  and  R  be a r i n g and x e R . C l e a r l y <x> = <x> i f 2 o n l y i f x e <x> and hence i f and o n l y i f t h e r e a r e i n t e g e r s i a, such t h a t x = S a.x . U s i n g t h i s c h a r a c t e r i K  B.  1=2  1  z a t i o n i t i s c l e a r t h a t homomorphic images o f &'  and t h a t i f  A  i s a r i n g w i t h an i d e a l  £'-rings a r e I n B. such"'that b o t h  - 68 -  A/B £'  and B  a r e i n e'  i s an e l e m e n t a r y r a d i c a l  3-4.2  A e t' . T h e r e f o r e , by 2 . 3 - 1 ,  then  class.  PROPOSITION: s'-ring w i t h o u t p r o p e r d i v i s o r s  A non-zero  i s an a l g e b r a i c f i e l d o f prime  of zero  characteristic.  Proof: Let  R  If 0 ^ x e R  sors of zero. a, 0  ..., a  v  be a non-zero  K  such t h a t  x = E  S'-ring without proper  then there a r e i n t e g e r s a.xi , hence  e  v  =  i "2  w - w) = (we„ - x)w = 0 , so  we  = w  so  e„  i-1 E a.x —  i s an i d e n t i t y f o r <x> . L e t w e R . x(e  K  I 2  -  v  divi-  Then  e w = w .  Similarily,  i s an i d e n t i t y f o r R . I f 0 ^ v e R  then  2  e  y  e <v> = <v>  so  f o r a l l non-zero  e  y  e <v>-v c Rv . T h e r e f o r e R / (0) , R  v e R ; so, since  R = Rv is a divi-  sion ring.' Let  e  be t h e I d e n t i t y  o f R . Then  <e> f  the  p  <2e> = <2e>  r i n g of integers since characteristic a l l non-zero  of R  i s a prime.  w e R , R  = <4e> . T h e r e f o r e t h e  Since  i s algebraic.  Theorem 2 on page 183 o f Jacobson  [11],  e = e e <w> f o r w T h e r e f o r e , by R  is a field. Q.E.D.  - 69 3 . 4 . 3 COROLLARY: If  (0) / R e ?/  then  R  i s i s o m o r p h i c t o a sub-  d i r e c t sum o f a l g e b r a i c f i e l d s o f p r i m e . c h a r a c t e r i s t i c . in particular, R  So,  i s commutative.  Proof: Let <x> = <x>  (0)  R e &'  = ... = <x>  and  = (0)  x e R .  so  If x  x = 0 .  N  Hence  = 0  then  6  rings  7  have no non-zero n i l p o t e n t elements so the c o r o l l a r y f o l l o w s f r o m 1.2.1 and 3 - 4 . 2 . Q.E.D.  e* <_ t'  Prom 2 . 1 . 3 ( i i i ) we know t h a t  = (£')* .  The f o l l o w i n g theorem p r o v i d e s a c h a r a c t e r i z a t i o n o f £* = t'  which makes i t c l e a r t h a t i n f a c t  £'  .  3 . 4 . 4 THEOREM: A ring  R e &'  i f a n d - o n l y i f e v e r y non-zero  f i n i t e l y generated subring of d i r e c t sum of f i n i t e  R  i s isomorphic to a f i n i t e  fields.  Proof: Assume t h a t  R e £'  f i n i t e l y generated subring of 3-4.3  R'  i s commutative.  and commutative  and l e t R' R .  Since  Then R  7  be a non-zero R ' e 5' so by  i s f i n i t e l y generated  we may c o n c l u d e , from the H i l b e r t B a s i s  - 70 -  Theorem, t h a t of  R'  3.4.2  R  s a t i s f i e s A.C.C.  7  and P' / R'  then  R'/P' i s a f i e l d .  commutative, and <g> of  R'  ;  i s a prime  <g.^>  ideal  i s a maximal i d e a l because by  Since  R'  i sfinitely  generated,  has an i d e n t i t y f o r each g e n e r a t o r  has an i d e n t i t y ( i f , f o r i = 1, 2 ,  R'  identity for  P'  I f P'  then  e  + e  1  2  - e^e^  e  g  i s an  ±  i s an i d e n t i t y f o r  <g ,g >).  Now-, by Theorem 2 , page 203 o f Z a r i s k i and Samuel  [15],  satisfies  2  R'  D.C.C.  Wedderburn r i n g so R' of f i e l d s .  Then  R'  i s a commutative  i s i s o m o r p h i c t o a f i n i t e d i r e c t sum  These f i e l d s must be f i n i t e , s i n c e t h e y a r e f i n i -  t e l y g e n e r a t e d and by 3 - 4 . 2 t h e y a r e a l g e b r a i c o f prime characteristic. The R'  converse i s o b v i o u s ; i n f a c t , i f x e R' and  i s i s o m o r p h i c t o a f i n i t e d i r e c t sum o f f i n i t e f i e l d s then  t h e r e i s an i n t e g e r  n ( x ) _> 2  such t h a t  x ^ ^= x . n  x  Q.E.D.  3.4.5  COROLLARY: A ring  R e &'  t h e r e i s an i n t e g e r  3.4.6  n ( x ) >_ 2  such t h a t  x eR  x ^ ^= x . n  x  COROLLARY: A ring  <x>  i f and o n l y I f f o r each  R e e'  i f and o n l y i f f o r a l l 0 4= x e R ,  I s i s o m o r p h i c t o a f i n i t e d i r e c t sum o f . f i n i t e f i e l d s .  If  ( 0 ) / R e e'  and R  has D.C.C. then  R  i sa  - 71  -  commutative Wedderburn r i n g so R d i r e c t sum o f f i e l d s i n Z' . that a condition sufficient  3-4.7  i s isomorphic t o a f i n i t e  I n t h e • n e x t theorem we see  which i s a p p a r e n t l y weaker than D.C.C. i s  to obtain this  result.  THEOREM: If  (0) / R e Z'  annihilators  then  R  and R  s a t i s f i e s A.C.C. on  i s I s o m o r p h i c t o a f i n i t e d i r e c t sum  of f i e l d s i n Z' ( t h a t i s , a l g e b r a i c  f i e l d s o f prime  characteristic).  Proof: Let  R  be a non-zero  By 3 . 4 . 3 R  A.C.C. on a n n i h i l a t o r s .  A.C.C. on l e f t a n n i h i l a t o r s annihilators,  R  . B  b  satisfies  i s commutative.  Since  i s e q u i v a l e n t t o D.C.C. on r i g h t  R = A^^ © . ... ffc A  are f i e l d s i n z'  z e r o elements  which  s a t i s f i e s D.C.C. on a n n i h i l a t o r s .  Assume  t h e n , by 3 - 4 . 2 ,  s'-ring  If  K + 1  2- 2 , b  €  h  a  s  n  R  $ B  o  P  r o  i s a f i e l d i n z' "^K+l  s  u  c  n  that  b  where t h e A  K + 1  P  e r  d i v i s o r s of zero  .  ]_* 2 b  I f t h e r e a r e non=  0  then  B^. ^  contains the a n n i h i l a t o r A  K + 1  = (0  : A  ©  ... © A © K  b-jR)  = t h e a n n i h i l a t o r o f A Q .. . © a © b^R / (0) 1  K  +  - 72 -  and  E  A K  +  1  B  K-fl  '  C  H  O  O  S  E  A  K+1  (°  =  :  C  K+1^  t  o  b  e  a  minimal non-zero a n n i h i l a t o r c o n t a i n e d i n B ^ ^ • Now, I f +  t h e r e a r e non-zero elements then  D = (0 :  such t h a t  A  K-KL  (  ^  Y  ty of  • Therefore  z e r o so  ^  an I d e n t i t y ,  i  a  s  a  K+2  '  T  H  E  R  E  F  K+1  O  R  E  A K +  i  s  B  ^  has no p r o p e r d i v i s o r s o f  B  +  by 3 ^ - 2 .  d i r e c t summand o f B  a  K  c o n t r a d i c t s the m i n i m a l i -  ^bis  2  K+2 ^  of R  a  B  n  d  K  +  R  Since K +  =  A  l®  " '  ^  A  i  +  h  a  s  - ^ . That-is,  such t h a t  1  N o t i c e t h a t t h i s p r o o f i s v a l i d when (0) 4 R  xy = 0  '  f i e l d i n &'  t h e r e i s an I d e a l B  such t h a t  + yR) i s a non-zero a n n i h i l a t o r (x e D )  p  D  x,y e  B  K  +  1  = •  A  K  +  1  @  ® K+1 © K+2 ' A  B  K = 0 , and s i n c e  we can b e g i n t h e above p r o c e s s . Since  R  s a t i s f i e s A.C.C. on a n n i h i l a t o r s , t h e  p r o c e s s above must s t o p .  That i s , f o r some  n , B  has no  n  p r o p e r d i v i s o r s o f z e r o and hence i s a f i e l d I n £' .  This  completes t h e p r o o f . Q.E.D.  T h i s completes o u r i n v e s t i g a t i o n o f t h e elementary radical class  s' .  We now p r e s e n t a c l a s s i f i c a t i o n o f a l l elementary r a d i c a l s which a r e  < PC .  3-4.8 DEFINITION: Define  t o be ft fl FC^ where  Jt i s any c l a s s  - 73 -  of r i n g s and p  3.4.9  i s a prime number.  PROPOSITION: 34  If  i s an e l e m e n t a r y r a d i c a l c l a s s and R i s  7  ( t t ' . n PC)(R) = © { » p ( R )  a r i n g then  : p  i s a prime} -  Proof: 34  Let be a r i n g . classes,  7  be an e l e m e n t a r y r a d i c a l c l a s s and l e t R  Since i n t e r s e c t i o n s of r a d i c a l classes are r a d i c a l M (R) and p  ( 34  7  n FC)(R)  are defined.  From 3 » 1 . 6 ( i i i ) we know t h a t f o r any r i n g FC(A) = © { F C ( A ) : p ( 34  7  i s a prime}  A ,  . So  n P C ) ( R ) = © ( F C ( ( 34 fl P C ) ( R ) ) : p 7  p  i s a prime}  . Now,  FC ( ( » ' fl F C ) ( R ) ) e 34 s i n c e 34 i s h e r e d i t a r y ; t h e r e f o r e , • ir FC (( 34 n. F C ) ( R ) ) c 34'(R) . S i n c e 34 (R) c ( 34 n PC)(R) and P P P Mp(R) F C , J l ( R ) c P C ( ( l i ' n F C ) ( R ) ) . Thus, 7  7  7  €  7  p  p  p  J4 (R) = F C ( ( 34 n p c ) ( R ) ) 7  p  7  p  . T h i s completes the p r o o f . Q.E.D.  3.4.10  DEFINITION: A s e t o f p o s i t i v e i n t e g e r s i s a C.U.D. s e t o f  i n t e g e r s i f and o n l y i f whenever i n t e g e r which d i v i d e s Suppose t h a t s e t o f i n t e g e r s and p  n , S  n e S  and k  i s a positive  k e S . i s a C.U.D. ( c l o s e d under d i v i s o r s )  i s any prime number.  Then f o r each  -  n e S  we c o n s i d e r  elements. divides  the f i n i t e f i e l d  Since n  74 -  P  S  then  F  n  = the f i e l d of p ^  I s a C.U.D. s e t , i f  n e S  k e S • hence, t h e s e t  and k  (F• : n e S) o f P  a l l such f i e l d s i s s t r o n g l y h e r e d i t a r y because a non-zero subring  of the f i e l d  F  i s the f i e l d  F ^ f o r some  P  divides  k  which  P  n . Let  ft  be t h e s e t o f a l l p o s s i b l e f i n i t e d i r e c t  sums o f t h e f i e l d s  F • : n e S . I t i s , i n f a c t , the c l a s s P  R  w h i c h we a r e d e f i n i n g i n t h e f i r s t p a r t o f t h e f o l l o w i n g  definition.  Notice  that the set of f i e l d s  (F ,  so as t h e C.U.D. s e t S  changes so does  of ft' on t h e prime  i s obvious.  p  R  : n e S} £ ft'  P  . The dependence  3 - 4 . 1 1 DEFINITION: (i)  ^(S)  i  s  t  h  c l a s s of a l l rings  e  R  with the property  ir  t h a t f o r a l l non-zero  x e R , <x>  i s isomorphic t o a  f i n i t e d i r e c t sum o f f i e l d s taken from  {F  : n e S}  P  where  p  i s a prime number and S  i s a C.U.D. s e t o f  integers. (ii)  3 7l(S)  i s the c l a s s of a l l r i n g s  perty that ring  R e FC  <x>/y\(<x>)  p  R  w i t h the  and f o r a l l x e R  pro-  the f a c t o r  i s i n IT (S) . 3?  I t i s c l e a r f r o m the above d e f i n i t i o n ^(S)  <L SpTK" ) • 3  I  n  t  h  e  that  f o l l o w i n g p r o p o s i t i o n we prove  these c l a s s e s a r e r a d i c a l  classes.  that  -  3.4.12  75  -  PROPOSITION: If  S  i s a C.U.D. s e t of p o s i t i v e  i s a prime number then radical  ^p(S)  and  3pT|(S)  I n t e g e r s and  p  are elementary  classes.  Proof: Let let  p  S  be a C.U.D. s e t of p o s i t i v e i n t e g e r s and  be a prime number. Prom the d e f i n i t i o n i t i s c l e a r t h a t  elementary c l a s s .  I t i s also  condition  (A)'. Suppose t h a t  and  both  B  and  A/B  and  A/B  are i n  e'  fields.  Therefore  B  are i n so  by 3 - 4 . 6 a g a i n , <x>  clear that  in  ^p(^) > ^  n e  ^p(S)  i s an  satisfies  i s an i d e a l of a r i n g  A  By 3 . 4 . 6 , b o t h  B  3 (S) .  A e &'  .  Let  0 / x e A .  Then  i s i s o m o r p h i c t o a f i n i t e d i r e c t sum of <x> fl B  i s a d i r e c t summand of  <x> = (<x>/<x> n B) Q> (<x> fl B) . are  3p(S)  Now because  A/B  <x>  so  and  B  f i e l d s i n q u e s t i o n must be o f the f o r m  P  where a e S . T h e r e f o r e , P 3" (S) i s a r a d i c a l c l a s s .  2-3.1,  A e 3 (S) ; s o , by P  P  Suppose  R e ^YKS)  and t h a t  R'  i s a homomorphic  sr  image of that  R .  <x'>  Let  x ' e R' .  i s a homomorphic image of  i s a homomorphic image of <x >/7i(<x >) e 3 (S) . ' p /  fore  Then t h e r e i s an  /  R' e Z 72( ) P s  s o  <x>/Yl(<x>)  Since ^TKS) P  R e PC  <x>  .  So  x e R  such  <x'>/Yl (<x '>)  , hence , R' e PC . p ' p  s a t i s f i e s condition  There(A) .  - 76 -  Suppose t h a t both  A/B arid  B  B  i s an i d e a l o f a r i n g  a r e i n 3" % ( S ) . B o t h  A  and t h a t  A/B and B a r e  3P  in  PC P  so A e FC P x e A and l e t  Let  <x>/yi(<x>) 2 . 2 . 6 , 'Yl(<x>)  7l(<x>)  3"p(S)  e  so  <x> = <x>/<x> 0 B . Then  is finitely  Thus, by  generated as a r i n g and s o ; s i n c e  i s a l s o n i l p o t e n t and i n P C ,  i s finite.  Yl(<x>)  p  Therefore  Now, by 2 . 2 . 6 a g a i n ,  <x> must be f i n i t e .  i s f i n i t e l y g e n e r a t e d as a r i n g . has  is finite.  <x>/Yl(<x>)  <x> fl B  <x> fl B/Yl(<x>  Since  n B)  no non-zero n i l p o t e n t e l e m e n t s , i t must be i n 3" (S) . 3?  Thus by 3 - 4 . ' 6  <x> n B / y K < x > n B) e t' so by 3 - 4 . 4  <x> fl B / ^ ( < x > n B) is finite.  Therefore, Let  <x> f l B + N N =  |  <x>/N  IVT«,T N o w  >  ~  N  S  <>  / < X >  X  .  N C E  <x> n B •  fl B  I T /—  =  , .  M  m  Yl(<x> n B)  =  L  <x> i s f i n i t e and  i s a d i r e c t summand o f <x>/N .  <x> n B + N ^ N  <x> i s f i n i t e .  i s a f i n i t e d i r e c t sum o f f i e l d s .  <x> n B + N  @  Now, j u s t as above, <x> fl B  N = yi(<x>) . S i n c e  commutative,  <|>  is finite.  +  N  '  N  T  {  <  U  X  > N  B  )  Let  ^  =  (  <  X  <x> n B + N  ,  H  ~ =  NN  S  '  N  >N  B  )  /  T  }  v  e  e 3 ( S ) •. p  <X>  M  <x> fl B + N •  Thus  T  H  . U  T S  L  ^  AT 1  • 3  A  homomorphic image o f <x> which has no non-zero n i l p o t e n t elements. <x>/fl(<x>)  <x>/M e  3  Hence e  L/N i s a homomorphic image o f  3- (S) p  . Thus  (S) .so A e  Z  yi(S)'  LA e y s ) .  . Therefore  - 77 -  From the d e f i n i t i o n i t i s c l e a r t h a t e l e m e n t a r y c l a s s so by 2 . 3 . I  3T ">?( ) s  l s  a n  p  3" 7 J ( S ) i s a r a d i c a l  class. Q.E.D.  3 (S) , ^ W )  The r a d i c a l c l a s s e s  s  a n d  p c  p  p  w i l l be our b a s i c b u i l d i n g b l o c k s f o r d e s c r i b i n g a l l e l e mentary r a d i c a l s ing  which a r e  <_ FC .  We b e g i n w i t h the f o l l o w -  result.  3.4.13  PROPOSITION: If  tt'  I S an e l e m e n t a r y r a d i c a l c l a s s and nVfc = ( ( 0 ) ) o r Yl fl F C  a prime then  p  is  < tt .  p  p  Proof: Let &  p  i f ' be an e l e m e n t a r y r a d i c a l c l a s s .  n 71 / ( ( 0 ) }  then t h e r e i s a non-zero r i n g  If  R e H  fl Yl  p  R e FC and R e Yi t h e r e i s an x e R , x / 0 p such t h a t x = 0 and px = 0 .' Thus <x> = C = the z e r o  Since  p  P  r i n g on the c y c l i c group o f  p  e l e m e n t s , so  C  e tt' .  P  A 6 71 n F C  Let  / (0) .  A = A/Bp (A) that  <x>  <x> e  Yl C\ F C y  and  pY =  .  p  = 0  and  2  tt  0  Let  .  So  Suppose  A  tt  .  p  A <t tt t h e r e i s an p  Then x e A  ( 0 ) / <x> = <x>/tt (<x>) . p  so t h e r e i s a  p  but .  Since  p  P  y / 0 .  Let  y e <x>  such t h a t  Y = (y) ~  i f w e Y , <w> «? Y  <  >  and  •  such  Now, py = 0  Then  Y  <w> = C P  2  = (0) e B' • P  .  - 78 Therefore so  Y e M  Y l n pc  p  p  . This i s a contradiction.  Hence  A e ft  p  < r . Q.E.D.  Suppose t h a t ft^ : a e A c l a s s e s such t h a t  and a l l r i n g s d i r e c t sum  » (R) n fi  i s a c o l l e c t i o n of r a d i c a l  S ft (R) = (0) f o r a l l j3 e A  R . Then f o r any r i n g  R  we can form t h e  (ft (R) : a e A} . I n such a s i t u a t i o n we s h a l l  denote by © {ft^ : a e A } t h e c l a s s o f a l l r i n g s which  R for  R = © ( » (R) : a e A } . I n terms o f t h i s n o t a t i o n P r o p o s i t i o n 3 . 4 . 9 t e l l s  us t h a t ft' n PC = © { f t '  "• p  i s a prime]  whenever ft' i s an  ir  e l e m e n t a r y r a d i c a l c l a s s ( r e c a l l t h a t ft I t f o l l o w s t h a t i f ft' < PC  p  = F C flft'). p  then ft' = 0 ( f t  i s a prime] . I n t h e f o l l o w i n g theorem we s h a l l prove o r 3^ Yl(S)  each ft^ must e q u a l e i t h e r  FC  some C.U.D. s e t o f i n t e g e r s  S . And c o n v e r s e l y ,  p  p  p  :p that  o r ^ (S) f o r any " d i r e c t  sum" o f such e l e m e n t a r y r a d i c a l c l a s s e s i s a g a i n an e l e m e n t a r y radical class. I n o t h e r words, e v e r y e l e m e n t a r y r a d i c a l c l a s s which i s c o n t a i n e d i n PC classes  i s a " d i r e c t sum" o f these s i m p l e r a d i c a l  (PC , 5" Yl(S) , 3" (S)) and a i l " d i r e c t sums" o f such  classes a r e elementary r a d i c a l c l a s s e s .  - 79 -  THEOREM:  3.4.14  F o r each prime  p , tt' = FC o r 3" f\(S ) o r P P P * p 3" (S ) f o r some C.U.D. s e t o f i n t e g e r s S i f tt' i s an P p' P e l e m e n t a r y r a d i c a l c l a s s . A l s o tt' fl FC = @ ( W p : p i s a prime} y  Conversely,  . tt' = © ^ [ ]  P  :  p  l s  an e l e m e n t a r y r a d i c a l c l a s s i f f o r each prime or  3 yi(S ) p ' ^ p'  or  S . Moreover, P  3" (S ) P P  a  P  r l m e  )  l s  p ,ttj.^ = F C  p  f o r some C.U.D. s e t o f I n t e g e r s  M = M , P [p]  f o r a l l primes  r  p .  Proof: L e t tt' be an e l e m e n t a r y r a d i c a l Define  class.  S = (n : F n e tt'} . Then P P  S P  i s a C.U.D.  s e t o f p o s i t i v e i n t e g e r s s i n c e tt' i s s t r o n g l y h e r e d i t a r y . We must show t h a t tt  p  p  fl tt' i s F C  p  ^(Sp) °  v  ;  I f tt' = FC P P We w i l l  consider If  we a r e done so suppose tt' / FC P P v  the two cases o f 3 - 4 . 1 3 -  »' n 7 l p  = ((0)}  Suppose t h a t  v  y  tt^ fl ft = { ( 0 ) } . L e t P  <x>/P ~ F [X]  <x>/P =.Fp[X]  Y\(<x>)  =  where  x  is a  since  (0)  be a prime i d e a l of  must have c h a r a c t e r i s t i c  f i n i t e or  we w i l l show t h a t tt' = 3" (S ). P p p  R e tt' = FC n tt' and t h a t P P  non-zero element o f R . Then  <x>/P  or  p  ) .  3 yi(s  p  = FC  <x> •.  p  so e i t h e r  X  i s an i n d e t e r m i n a t e .  Then  <x>/P I s  then e v e r y r i n g o f c h a r a c t e r i s t i c  p  If  is in  M' ( s i n c e t h e y a r e a l l homomorphic images o f F [ X ] ) .  - 80 -  But then that  FC  <_ ft' because i f A  p  A/B e ft' and B e ft' then  I s a r i n g and B <5 A A e ft' . Hence ft p  <x>/P  i s finite.  Then  <x>/P  z e r o d i v i s o r s so i f P / <x> hence  P  .  p  p  Hence  i s a Wedderburn r i n g w i t h o u t  then  <x>/P  i s a f i e l d and  i s a maximal i d e a l o f . <x> . S i n c e  A.C.C. and a l l prime i d e a l s a r e maximal, <x>  <x>  satisfies  s a t i s f i e s D.C.C.  (Theorem 2 , page 203 o f Z a r i s k i and Samuel [ 1 5 ] ) . <x>  = FC .  p  T h i s i s c o n t r a r y t o o u r s u p p o s i t i o n t h a t ft / F C  such  Therefore  i s a commutative Wedderburn r i n g so  <x> = F a, © ... © F a  . Then  v  p  l  p  P a . e ft' so t h e a. e S  K  p i  R e 3" (S ) . P P  Therefore  Conversely, I f R e ^ p ^ p )  t  l  h  e  n  f  o  r  a  1  p  x e R ,  1  <x>  i s a f i n i t e d i r e c t sum o f f i e l d s i n ft' so <x> e ft' . P P T h e r e f o r e R e ft' , so ft' = 3 (S ) . P ' P P P I f J! we w i l l show t h a t tt'p=p 3" *W(S uv, n PCp < _ ft' p p ). 5  Suppose above  <x>/Jl(<x>)  V  ;  R e ftp = PCp n ft' and x e R . Then as e  ff  (S )  p  p  . Since  R e ft , R e F C p  p  so  R e ? H(S ) . p  p  Conversely, i f R e 3 T l ( S ) p  <x>/*|i^(<x>)  p  then f o r a l l  x e R ,  i s a f i n i t e d i r e c t sum o f r i n g s i n ft' so 3?  <x>/y|(<x>) e ft^ . S i n c e Yl(<x>) e P C Hence ft  p  p  R e 3 Yl(S ) , R e PC p  f\fl < ft . T h e r e f o r e p  = %Yl(S ) p  .  p  p  so  <x> e ft^ so R e ft . p  - 81 -  Prom 3 ' 4 . 9 we know t h a t 34' n FC = © [ M p  ': p  i s a prime} .  . We s h a l l now prove the c o n v e r s e . 34 = Q {Jij- -j : p  i s a prime}  p  or  3" (S ) p p  prime ti <  ivhere  Assume t h a t  34j- j = F C p  f o r some C.U.D. s e t o f i n t e g e r s  p .  Since  w  [p]^  ^  L  5  p  S P  f ° --- P i r a  o r 3 "y2(S  p  r  f o r each s  m e  p ,  FC . F i r s t we s h a l l prove t h a t ti = ti' . Suppose  and l e t x e R . S i n c e prime} . T h e r e f o r e  R e ti , R = © [34j- j(R) : p  i sa  p  x = x  1  + ... + x  where  n  x  i  e 341-  a.  and  P  1 i  ( x  x = 0  f o r some I n t e g e r s  ±  such t h a t  i)<x>  -j (R 1  a  d.x = x. . S i n c e i i  . . i <x>  0  S  ( x  )  =  *  ( 0 )  jVi  <x> = ( x ) 1  + ... + ( x )  < x >  hereditary, (  n  x i  )  e < x >  T h  > 1 .  ±  i ¥ 3 , Pj_ ¥ P j • Thus, f o r each d. i  R e  Moreover, i f  i , t h e r e i s an i n t e g e r  v  ( x . ) . e'FC i <x> p ;  ,  5  v  i  erefore, . Since  < x >  H[ j  i s strongly  p  **[p.] ' T h e r e f o r e ,  <x> e 34 ; so  R e 34' . Suppose R = © {FC (R) : p p  FC (R) = M[ ](R) • p  p  x e FC (R) p  .  Since  <x> = © {34j- -j (<x>) p  R e 34' .  Then  R e PC  so  i s a prime} . We s h a l l show t h a t Clearly  M  ( ) £PC (R) . Let R  [ p ]  R e 34' , <x> e 34 . : p  p  Therefore,  i s a prime} . B u t  <x> e F C  p  so  - 82 -  M  [ ] ( < > ) = (0.) f o r a l l q / p  . Thus,  x  q  <x> e tt  . Since  [p]  FC (R) = [ ] ( ) W  p  mentary c l a s s .  Suppose  Therefore,  Hence tt = tt' i s an e l e -  so c l e a r l y  )  :  J  Rett  R . Then R /  .  [p]  so  N o t i c e t h a t we have shown t h a t r  R' = © ^ [ p ] (  p  p  Rett.  tt = FC flttc tt , P P ~ LPJ  image o f  M  = tt'^ , F C ( R ) e tt  [p]  so  R  p  tt  <x> = [ ] ( < x > )  M  =Ttt , . r  P  Lp]  and l e t R' = R/I  [ ]( ') - ^[p]^) R  be a homomorphic  +  s 0  c  l  e  a  r  l  p  i s a' prime} . T h e r e f o r e  p  y  R' e tt so  tt s a t i s f i e s c o n d i t i o n ( A ) . Suppose and  A  i s a r i n g and B < A  B e tt . S i n c e b o t h  Therefore,  A/B  p  a r e i n FC , A e FC .  i s a prime} .  p  tt|- -j  B  A/B  by 3 . 1 . 6 ( i i i ) A = © (FC (A) : p  Since  and  such t h a t  B e tt , F C ( B ) = [ ] ( B ) W  p  p  (*)  and s i n c e b o t h  FC  and  p  are h e r e d i t a r y (see Theorem 4 8 i n D i v i n s k y [ 7 ] ) , tt (A) [p]  n B =  «  [  p  ]  ( B )  = F C ( B ) = F C ( A ) fl B . p  (**) .  p  Now ( F C ( A ) / F C ( A ) n B) = ( F C ( A ) + B/B) c P C ( A / B ) e p  p  p  p  tt  [p]  since  A/B e tt . By ( * * ) , F C ( A ) D B e tt^j , so s i n c e  tt^  radical class,  EFC (A)  p  F C ( A ) e ttj- -j . p  p  Clearly  **[ ]( ) A  p  is a p  - 83 -  so  M  [ ]( ) = A  F C  p  ( )  •  A  p  N  o  (*) i m p l i e s t h a t  w  T h e r e f o r e , by 2 . 3 . 1 ,  ft  A e ft .  i s an e l e m e n t a r y r a d i c a l  class. Q.E.D.  The f o l l o w i n g l i s t p r o v i d e s a r e p r e s e n t a t i o n o f each o f t h e e l e m e n t a r y r a d i c a l c l a s s e s ft <_ FC have a l r e a d y d i s c u s s e d .  Let Z  which we  be t h e s e t o f a l l p o s i t i v e  integers. =© = 0  (3p(.Z ) : p  i s a prime] .  +  £>' = W n FC = Q {X 71(0) P C  p  =  ^ (  z  +  :P  i s a prime] .  ) •  71 n p c = ^71(0) • p  <% n FC =Q  {3 72(Z ) +  p  :p  i s a prime] .  The r e l a t i o n s h i p s between t h e e l e m e n t a r y r a d i c a l c l a s s e s w h i c h we have d i s c u s s e d can be i l l u s t r a t e d by t h e f o l l o w i n g diagrams.  - 84 -  ILLUSTRATION 1  - 85  -  CHAPTER I V GENERALIZED ELEMENTARY AND LOCAL RADICAL CLASSES  4.1 ABSORBENT CARDINAL NUMBERS. I n t h i s f i r s t s e c t i o n o f Chapter I V we s h a l l prove t h a t 2 and  |^  are absorbent c a r d i n a l s .  show t h a t any c a r d i n a l absorbent.  K  such t h a t  I n f a c t , we s h a l l  2 <_ K <_ f-f  is  We b e g i n w i t h t h e f o l l o w i n g two lemmas.  4.1.1 LEMMA: If  R 4 (0)  characteristic  p  and  p  i s a zero r i n g  f o r some prime  m o r p h i c a l l y mapped onto group o f  R  p  then  R  ( R = ( 0 ) ) of 2  can be homo-  Cp = the z e r o r i n g on t h e c y c l i c  elements.  Proof: Let pR = (0) . K  and  be a r i n g such t h a t  Choose  Then,  such t h a t  x e (w) + K R  x = nw + y  x ^ K , p  integers  r  and  R / (0) , R  2  = (0) and  x e R , x / 0 . By Zorn's Lemma choose  maximal i n Z = ( 1 4 R : x | 1} .  w k K . n  R  when  such t h a t  0-/-w e R  so t h e r e i s a non-zero y e K .  does n o t d i v i d e s  Suppose  n .  Since  and  integer  pR = (0) c K  Therefore, there are  r p + sn =. 1 ; s o ,  w = ( r p + sn)w = snw = sx - s y .  Thus,  w e <x> c R/K  so  R/K = <x> =• Cp . Q.E.D.  - 86 -  Let  p  be a prime number.  cussed i n Rings and R a d i c a l s . D i v i n s k y  The r i n g [7] •  p  i s dis-  We may t h i n k o f  t h i s r i n g as t h e s e t o f a l l r a t i o n a l numbers o f the form — n where p ^ does n o t d i v i d e a . The a d d i t i o n i s modulo 1 P and t h e m u l t i p l i c a t i o n i s t r i v i a l . A l l i d e a l s of p are n n i s o m o r p h i c t o Cp = t h e z e r o r i n g on t h e c y c l i c group o f p elements f o r some p  n , and a l l non-zero homomorphic images o f  are isomorphic to p  ra  M  .  4.1.2'LEMMA: If that  x / 0  R  i s a zero r i n g and t h e r e i s an  b u t px = 0  f o r some prime  homomorphically mapped onto  Cp  o r onto  p p  x e R  then ra  R  such can be  .  Proof: Let x e R I  R  such t h a t  be a r i n g such t h a t x / 0  R  = (0) and l e t  b u t px = 0 .• By Zorn's Lemma choose  maximal i n Z = ( I < s Q R : x < ^ I } . If  p ( R / I ) / R/I then  R/I  can be homomorphically  mapped onto a zero r i n g o f c h a r a c t e r i s t i c 4.1.1  R  p  can be homomorphically mapped onto Suppose  p ( R / I ) = R/I . L e t w e R  Then  x e (w) + 1  that  x - nw e I . T h e r e f o r e  R  so by Lemma Cp . and w £ I .  so t h e r e i s a non-zero i n t e g e r  n  such  pnw e l so R/I e FC . Now  FC ( R / I ) i s a d i r e c t summand o f R/I and hence a homomorphic  - 87  image of  R/I  K  and s i n c e  of  R  t y of  .  I , FC  w e R/I  where  p ( R / I ) = R/I such t h a t  p  such t h a t  are Integers w = (rp  and  Now,  n  be a p o s i t i v e i n t e g e r . n  .  —  z  p w = 0 .  Since  r  such t h a t  and  s  R/I  the i d e a l s of  the a b e l i a n group  v e R/I  so t h e r e i s an i n t e g e r  p  does not d i v i d e rp  k  + sn' = 1  n .  + sn )w = sn w = sn p v = n p ( s v ) = n ( s v )  = (0)  R/I  +  .  is divisible.  +  R/I  there Then  .  There-  Since  are j u s t the subgroups of  T h e r e f o r e , by the theorem f o r  d i v i s i b l e t o r s i o n groups (see f o r i n s t a n c e , Puchs [ 8 ] ) , Is i s o m o r p h i c t o a d i r e c t sum R/I  Write  Since  so t h e r e I s an element R/I e FC  f o r e , the a d d i t i v e group (R/I)  maximali-  .  a  = w .  f o r some i d e a l  (R/I) , x £ K .. By the  , p ( R / I ) = R/I  p°v  (R/I) = R A  does not d i v i d e  • ie—  k  PC  x e PC  (R/I) = R/I  Let n = p n  Therefore,  -  of c o p i e s of  p°° .  R/I  Therefore  can be homomorphically mapped onto the r i n g  . Q.E.D.  We a r e now  4.1.3  THEOREM: If  K  Is  r e a d y to prove the theorem.  K  i s a c a r d i n a l number and  2 <_ K <_ |^  then  absorbent.  Proof: Let  K  be a c a r d i n a l ,  2 <_ K <_ j$  .  Suppose t h a t  - 88 -  I + IR + RI + RIR Suppose t h a t homomorphic image  M £ J  M' = M/K'  F i r s t we s h a l l prove t h a t Let  x e I . then  mx =  N S  ir=i  i s a non-zero I n t e g e r . 2 such t h a t (km)x e I nw e M  3  3  such t h a t no K - s u b r i n g o f  1  where  Suppose  I / I e FC .  u. , v. e I 1  1  and  Now, t h e r e I s a non-zero i n t e g e r  3  L  m  1  k  .... n . T h e r e f o r e  e FC . B u t then i f w e M = ( l ) , R  and  n / 0 . This i s impossible  <z> \ FC .  Therefore  I / I £ FC  so  can be homomorphically mapped onto a non-zero r i n g = (0) and  L  L  has c h a r a c t e r i s t i c 0 ( f a c t o r  FC(I/I )). 2  x e L , x 4 0  maximal i n the c l a s s  i n t e g e r then y \. H  but  maximality k  u.v.  f o r some i n t e g e r  Choose H  Then t h e r e i s a non-zero  I / I | FC -  for 1 = 1 ,  I/I  z e M/K'  such t h a t out  , so  c K'  because 2 I/I  ku. e I  g(K) of M  so  nx ny e H  of  J} .  Z = {J<JL:if Then  n  i s a non-zero  FC(L/H) = (0) , f o r I f y e L,  f o r some non-zero i n t e g e r  H , kx e ( y ) ^ + H  (nk)x e H  and by Zorn's Lemma choose  n  then by the  f o r some non-zero i n t e g e r  which c o n t r a d i c t s t h e way i n which  H  was  - 89 -  chosen. Let S  S  be a K - s u b r i n g o f  Is f i n i t e l y generated.  ideals  of  Since  S  S  L/H .  Because  L/H  Since  K <_ ^  ,  i s a zero r i n g t h e  a r e j u s t the subgroups o f the a d d i t i v e group  i s f i n i t e l y g e n e r a t e d we may a p p l y  S'  the fundamental  theorem f o r f i n i t e l y g e n e r a t e d a b e l i a n groups t o see t h a t i s i s o m o r p h i c t o a f i n i t e d i r e c t sum o f c o p i e s If  u  and  v  U  integers  H  + L  r  a  n  d  k  and  n  nx e ( v ) ^ + H .  and  s  such t h a t  i s of l e n g t h 1 ; that i s ,  Hence t h e r e a r e non-zero  kx - r u e H  and  L/H  I e $ g(K) '  s  o  m  e  paragraph  K-subring of  I ) i s i n el  are Isomorphic to  C  i s a c o n t r a d i c t i o n s i n c e we assumed t h a t contain a  nx - s v e H.  S * C°° .  (which i s a homomorphic image of K-subrings of  L/H  such t h a t  T h i s i n s u r e s t h a t the d i r e c t sum i n t h e p r e c e d i n g  Now, s i n c e  C°° .  a r e two non-zero elements o f  t h e r e a r e non-zero i n t e g e r s kx e ( )  of  05  .  Since a l l  , 0°° e o$ M/K  7  L/H  .  This  d i d not  K-subring.  Case 2:  There i s a  f o r some prime If  z e M  such t h a t  Cp = <z> c  MA'  p . 2 2 p(l/I ) / I/I  then  2 I/I  can be homomorphi-  c a l l y mapped onto a zero r i n g L / (0) o f c h a r a c t e r i s t i c 2 By Lemma 4.1.1, I / I can be homomorphically mapped onto Cp .  S  So, s i n c e  I e «5 S  , Cp e J (K)  .  p.  This i s a c o n t r a d i c t i o n  - 90  -  because we have assumed t h a t  R  x. e I l  v., s., v., s. e R . i ' i ' i ' l  and  Now,  p ( l / I ) = I / I , so 2  2  n .  3  .  Thus  F o r each  x. - p n x. e i ^ I n  T  2  + I  n  3l  n  3  choose  =  .  L E  ,  1=1  3  Moreover, s i n c e M/K  3  7  (*) v  = PI + p i  2  for a l l positive  x.  e l  so  n  n  2  + I  3  integers  such t h a t  '  z" = p z" "z  /  Hence  r . x . s. + r . x . + x. s. + 1 1 1 i i i i n n n  pz = 0 , p  7  = I .  l n  e M <= K  n  2  I = pi + I  .  z  z - p z  so  3  p (l/I ) = I/I n >_• 1  Let Then  pi + I  2  = I(pl+ I ) = p i  pi + I  such t h a t  L _ _ S r . x . s . + r . x . + x . s . + m.x. . i l l i i i i i i  z =  2  has no K - s u b r l n g s i n  7  2 2 We must conclude t h e n , t h a t p ( l / I ) = I / I z e M = ( l ) t h e r e are i n t e g e r s nu and elements  Since  I  M/K  = 0  / 0 so  in  m.x. i  M/K  i  n  .  .  7  < > = Cp  in  z  n  Now,  suppose t h a t f o r a l l x e I , x e ( p x ) + I 'I Then i f x e I , t h e r e i s an i n t e g e r n such t h a t 2 2 2 x - pnx e l so ( l - pn)x e I . T h e r e f o r e I / I e FC . T  Suppose  FC (I/I ) 2  p  t h e r e i s a non-zero i n t e g e r 2 r and r x = Y. u.v. e I . 1=1  = (0) . r  Let  such t h a t  x e I . p  Then  does not d i v i d e  H  1  s  1  which i s not d i v i s i b l e by  1=1,  H .• Thus,  Now  t h e r e i s a non-zero  integer  p p  srx e I  such t h a t and  p  su^ e I  for  does not d i v i d e  - 91 -  sr  . Therefore,  we may choose a non-zero i n t e g e r  i s n o t d i v i s i b l e by  p  and such t h a t  kx^ e I •  i = 1, ..., L . Now from ( * ) , k z e ( l )  £ M  3  c o n t r a d i c t s o u r assumption t h a t Therefore I/I  2  R  for c K' .  This  F C ( l / I ) / (0) so by Lemma 4.1.2 2  p  05  can be homomorphically mapped onto  x £ (P )I  t  which  <z> = Cp .  Cp  X  mapped onto  +  I  2  p  such  2  I / I can be homomorphically  then  2  I/((px)j + I )  homomorphically mapped onto  o r onto x e I  On t h e o t h e r hand, i f t h e r e i s an tha  3  k  which by Lemma 4.1.2 can be Cp  o r onto  p" . 5  Thus o u r a s s u m p t i o n i n Case 2 leads t o t h e c o n c l u sion that  I  can be homomorphically mapped t o Cp  •gT . We have seen t h a t t h e c o n c l u s i o n t h a t m o r p h i c a l l y mapped t o Cp Now, i f I P  CO  Cp  leads to a  I  or to  can be homo-  contraduction.  can be homomorphically mapped onto  p , ra  e 520 ~ • Since a l l K-subrings a r e f i n i t e l y generated, S(K) e  n  f o r some p o s i t i v e i n t e g e r  <z _ > e n  i f n _> 2  1  n . B u t then  and <z> e g(f  i f n = 1 . I n any M / K ' c o n t a i n s no  case t h i s c o n t r a d i c t s o u r assumption t h a t K-subring i n  E i t h e r Case 1 o r Case 2 must o c c u r s i n c e (M/K'p  =  (0) . B o t h cases l e a d t o a c o n t r a d i c t i o n so we  conclude t h a t  M e J(  g(K)  . Therefore  K  i s an a b s o r b e n t  cardinal. Q.E.D.  - 92 -  ^•2 GENERALIZED RADICAL CLASSES WHICH ARE  4 fj  . O  We s h a l l r e f e r t o g e n e r a l i z e d 2 - c l a s s e s as g e n e r a l i zed e l e m e n t a r y c l a s s e s .  Generalized  V ^ - c l a s s e s w i l l be r e -  f e r r e d t o as g e n e r a l i z e d l o c a l c l a s s e s . f o r ti i it \  and  ti  f o r ti  , .  (  We s h a l l w r i t e ti o  The f o l l o w i n g p r o p o s i -  t i o n shows t h a t t h i s w i l l n o t c o n f l i c t w i t h our n o t a t i o n f o r the g e n e r a l i z e d n i l r a d i c a l c l a s s o f A n d r u n a k i e v i c and Thierrin.  4.2.1  PROPOSITION: o  Proof: Assume t h a t  R e ~fl. • L e t R'  be a non-zero  CD  homomorphic image of R . Then  R' e "Y?  so by 1 . 2 . 1  there  S  i s a non-zero n i l p o t e n t element i n R' . T h e r e f o r e R e *^ (2)  If  g  R eY? g(2)  then e v e r y non-zero homomorphic  image o f R c o n t a i n s a n i l s u b r i n g so c l e a r l y no non-zero homomorphic image o f R i s fl s . s . T h e r e f o r e , R e 77 g g By 2 . 4 . 2 ( v i i ) ,  7^(2)  =  ) so  Q.E.D.  4 . 2 . 2 THEOREM: If  ti  i s any c l a s s o f r i n g s such t h a t  j3 <_ti<_ FP  - 93  then  tt  g  =  -  Y\  §  1  Proof: Assume t h a t Since  *y| =71 g  > Tig  g-j^  and  tt  i s a c l a s s of r i n g s and  £ <_tt-<_ FF,  , by 2 . 4 . 2 ( i i ) we need o n l y show t h a t  FF  < 7?  g  '  S i n c e a r i n g g e n e r a t e d by one element i s i n and o n l y i f i t i s i n 71  I t i s clear that  = Yl  6 s  Let  R  l  R £ Y(  be a r i n g such t h a t  j3 i f  = 71- • l . Then t h e r e s  .  s  g i s a non-zero homomorphic image  R'  has no non-zero n i l p o t e n t e l e m e n t s . 0 / x e R'  , <x>  i s 71  p r o v e d t h a t such a r i n g  of  such t h a t  R'  Thus, f o r a l l  semi-simple. <x>  R  I n Theorem 3 « 3 - 2 we  c o u l d be homomorphically mapped  onto a f i n i t e f i e l d . Hence, no non-zero s u b r i n g i s i n FF , so R I PP . T h e r e f o r e , FP < g-L g ~  <x>  of  R  7  2  Q.E.D.  R e "Yl  If of  R  t h e n e v e r y non-zero homomorphic image  contains a subring  Thus, R e jS  <_ jS  g-^ yi  must have  'S L  <x>  such t h a t  (by 2 . 4 . 2 ( i i i ) ) .  Now,  <x>  i s nilpotent.  by 2 . 4 . 2 ( 1 1 )  we  g = «£ = j3 g g  U s i n g 2 . . 4 . 2 ( i i ) a g a i n we see t h a t 7) y i  < J < g $ < 3 ' < F < F F . g ~ g g - g - g g J  Almost a l l q u e s t i o n s con-  c e r n i n g t h e s e r a d i c a l c l a s s e s a r e open. which of the above i n c l u s i o n s a r e s t r i c t .  We do not even know N o t i c e however,  -  that  FF  < PP g -  9k  -  since subrings  of f i n i t e f i e l d s a r e f i n i t e  b  fields.  The next p r o p o s i t i o n i s concerned w i t h the g e n e r a l i oSr  zed c l a s s e s a s s o c i a t e d w i t h 2.4.2(vii)  4.2.3  &  =  g  *  and'  and (& ) R  %^  •  N o t i c e t h a t by  = (& )  g  •  R  PROPOSITION:  (i)  R e &  (ii)  * .  (iii)  R  i f  o  n  l  y  R / ( & ) ( R ) e PC •  i f  R  * C f r  g  a n d  g  R  )  g  *  Tig  g  •  is  %-  s e m i - s i m p l e i f and o n l y i f f o r a l l x e R,  =  fax]  •  <x>  Proof: (i)  Assume t h a t I O R  Re  &  such t h a t  Since  R  for  such t h a t  ma x n  n  R / ( & ) _ ( R ) $ FC R  m  (R)  and  a^,  £  a  x  n  n  a  n  .  g  let  I / ( # ) (R) =  and  R/I e  x £ I  some i n t e g e r s  zero i n t e g e r  If  I = (&v)  FC(R/( & ) ~(R)) • f g x e R  .  R  t h e r e i s an  + ••• + a^x e I Then t h e r e i s a non-  such t h a t  + ... + ma x e ( & „ ) 1 Rg n  v  /  x  ( R ) .. '  Let  and l e t y = ma x . Then n n . . . n-1 n-l + ... + b € (l ) (R)  b. = ma. (ma ) I 1. rv'  n _ 3  J  y  +  b  y  i  Since x £ I , y ^ I  y  R  g  so c e r t a i n l y  •  (f)  y <t ( & ) R  (R) .  We  - 95  -  _ <y> + ( & ) , ( R ) <y> = — ( vg. ) ( R )  By (j>), t h e a d d i t i v e group g e n e r a t e d so I f  w € <y> , <w>  g e n e r a t e d a b e l i a n group. f (w),  f (w)  1  h  b^,  .  l n  R  is finitely  +  i s also a f i n i t e l y  Thus t h e r e a r e p o l y n o m i a l s <w>  .  +  Choose an  which i s l a r g e r than the degree o f each h  f.  i s  <y>  which generated  K  integer  ^ ( Jr )g-  R  w i l l prove t h a t  L  Then  w  =  - b g  .  Therefore, Jjr-^  Since <y> e ( X*R)g  S b.f.(w)  •  f o r some i n t e g e r s <y> e  s a t i s f i e s condition (A),  Now s i n c e  f>  4 . 1 . 3 ) the non-zero i d e a l of generated by  <y>  contradiction. Both  Hence  G  i s a b s o r b e n t (see which i s  •  This i s a  (R) e PC .  R  (  Q  R  R/( & )  and  c  R / ( $ ) (R)  ( ^R)  is in  PC  R  „  are  <_ %• g  g  so the  converse i s o b v i o u s . (II)  Clearly % > (& ) but i s not i n ( & ) g  R  g  R  f i e l d i s i n y\  > )t • so & g  g  The r i n g ¥ (** ) -  b u t they a r e a l l i n  (& ) ¥ yi •  (iii)  R  P [X] e finite p  N o  g  ( fr-n)  R g g T h i s f o l l o w s i m m e d i a t e l y from 2 . 4 . 1 5 and  ^  SO  3.1.3Q.E.D.  We s h a l l now c o n s i d e r the c l a s s e s  ((&- ) )' R  and  - 96 -  (  )' . By 2 . 4 . 2 ( i x ) b o t h c l a s s e s a r e r a d i c a l c l a s s e s and  clearly  ( ( & ) ) ' = (( £ ) ) * R  g  R  Since  $y and  &  ( \Y  >  &  and  If  R e ( ^- ) ' g  g  g  ((& ) )' > % R  g  R  •  and x e R  then  <x> 6 jjr so R e tr  <x> e  •  ^  o  so  Therefore,  •  =  Now  4.2.4  (& )' = (£ ) * .  a r e elementary r a d i c a l c l a s s e s /  R  <x> % ( p [ X ] , thus ( &g)  and  g  £  R  < ( *  )  R  < &  g  so  g  & <((& ) )'< R  R  g  PROPOSITION:  P C n ((& ) )' < ^  (1) (2)  R  g  R e ((& ) )' R  R  i f and o n l y I f ' f o r a l l x e R ,  g  has c h a r a c t e r i s t i c 0. and i s i n *tr  <x>/J» (<x>) R  Proof: (1)  Suppose  R e PC fl ( ( &  i s a prime]  R  )  so s i n c e  ) ' . Then efr  s u f f i c i e n t t o show t h a t prime  i s a radical class i t i s  R  PC  p . Let A e PC  R = £ ) ( P C (R) : p  fl  p  (  (  «  f  r  R  )  g  )  fl ( ( & ) ) '  p  R  '  <L £ - ' f o r each R  and l e t x  g  be a  non-zero element o f A . Then t h e r e a r e i n t e g e r s a^,  and a  that b  K-l'  a^  y = a^x + ... + a^x  K K-1 y + b _^y + ... + b^y e p<x> K  ''''  . Thus  b  l'  W  e  m  a  TT }  x^ = x  y  a  s  s  u  m  e  + ^Yil-1  that XT 9 _1 x  p  p<x>  such  f o r some i n t e g e r s  does n o t d i v i d e  + ... + c x e p<x> n  -  f o r some i n t e g e r s above argument  97 -  c^, . ..,  c  K-t-i *  B  r e p e a t i n g the  y  we see t h a t f o r each i n t e g e r  t h e r e i s a monic p o l y n o m i a l  in x  n _> 1  which i s ' i n  p <x> n  Ci  Since  p x = 0  Assume t h a t <  > / c t ?  x  R  ( <  x  i s $y  < t  /  > <*>  <x>/e1r (<x>) e i r such t h a t  a^  y  + K-l  K  a  R  K >_ 2  of t h e a^ y  <x>  -  1  +  0  we may  and t h a t t h e g r e a t e s t common d i v i s o r  ^ K K-2^ ~ a  f o r integers  has c h a r a c t e r i s t i c  i s 1 . Now i f  K  0 . Let  <x> = <x>/^ (<x>) / (0) . Then  Since  assume t h a t  Since  assume t h a t f o r a l l x e R ,  so a^jc^" + . . . + a-^x = 0  a^,  0 .  and has c h a r a c t e r i s t i c  R  <x> e I-  by p a r t ( l )  € tr •  Conversely,  x e R  semi-simple,  R  must be o f c h a r a c t e r i s t i c  R  g  x e R . Then s i nee  R  <x>/<£ (<x>) R  a , we- conclude t h a t  R e ( ( J r ) ) ' and l e t  > )  ((^ ) )  f o r some i n t e g e r  a  y K  T h i s guarantees t h a t  y = a^x , +••••+ ( K  2  a  <y> e ( $ ) R  K _ 2 a  l^  y  =  0  . Therefore,  ' since  O  i s absorbent, a  K  a  x  =  y  ^ ^<x> y  a  n  d  <  t  n  i s 1 , the r i n g  i  (& ) R  so  e  (y) ~  R  • Thus,  g  e  ((tr ) )' R  s  e  S < x >  r e a  e (£ )  >  R  g  • Now, s i n c e  t e s t common d i v i s o r o f t h e  /(y)<x>  <x> e ( £ - ) R  g  i s  f  i  n i t e  • Therefore,  and hence i n <x> e (J&- ) R  .  Q.E.D.  - 98 -  I t f o l l o w s Immediately t h a t  ((Ct ) )')  By 2 . 4 . 2 ( v i i )  4.3  R  g  g  Jj-  = (£ ) • R  g  GENERALIZED RADICAL CLASSES WHICH ARE 4.3.1  R  4 FC .  LEMMA: Let  p  c a r d i n a l numbers  be a p r i m e .  Then  (  F C p  ) (K)  =  F C  g  p  f o r  '  a 1 1  K >_ 2 .  Proof: Let  p  be a prime and K  Since  PC P strongly hereditary, Suppose R -/ (0) t h e n s i n c e subring  S  of R  be a c a r d i n a l  >_ 2 .  s a t i s f i e s c o n d i t i o n (A) and PC i s P P C <_ ( )g( p c  p  p  K  R e ( C ) ( )  . L e t R = R/PC (R) . I f  F  p  g  )*  K  p  R e (FC ) t h e r e i s a non-zero KP g(K) such t h a t S e F C . B u t then ( S ) ePC R  p  which i s a c o n t r a d i c t i o n s i n c e _ Hence R = (0) so R e FC ' P  R  i s PC P  semi-simple.  v  Q.E.D.  4.3.2  PROPOSITION: L e t tt be an e l e m e n t a r y r a d i c a l c l a s s  let  M= ©(W  : p e S}  be t h e r e p r e s e n t a t i o n o f tt g i v e n  XT i n Theorem 3• 4.14.  <_ FC and  Then f o r any c a r d i n a l  K >. 2 , R 6 ^o-(K)  p  -  i f and o n l y i f R = © ( ( denote  tt ^  by © C (  g  M  99  -  ) ( )( )  : P e S) .  R  p  M p  g  K  ) ( ) g  Thus we may  : P e S] .  K  Proof: Let and  let K  S  be a s e t o f p r i m e s ,  be a c a r d i n a l  (W ) / \ < (PC ) t v \ = p'g(K) p'g(K) Tr  v  p  (  M  p  ) (K)( ) R  p  R e  M g  i n t h e p r o o f of 4 . 3 - 1 , R =©(PC (R) : p  image o f , s g(K) Tr  H e tt . 4  ' ' 3  1  R =Q  Then  K  (  R  R  g  (K)  F  C  C(»p)g( )( ) R  K  p :  3  P  e  0  P.—< tt  a  n  d  J  '  u s t  a  s  p ,  R e ^ ( K )>  so s i n c e  be a non-zero homomorphic  i s a homomorphic image o f  H e tt fl FC = W p P  Vg(K)^  tt  P C  Thus, R = © (FC (R) : p € S] .  • p F C  ( R )  so .  =  (  H c R  such t h a t  PC (R) e (U) , . . By P P g(K)  Vg(K)  '  ( R )  T h e r e f o r e  >  S) •  C o n v e r s e l y , suppose t h a t Since  R e  so t h e r e i s a non-zero K - s u b r i n g Now  ,  p  R , t h e sum  Now, f o r any prime  and l e t R  FC (R) . Then  <_ F C  p  Hence by 3 - 1 - 6  R e FC .  p  Rett  Thus, f o r any r i n g  ( )•  P C ( R ) = (0) i f p <t S . p e S  = tt n P C  p  i s a homomorphic image o f  Let  p  S i n c e tt  i s a prime] .  p  ( t t : p e S] ,  direct.  i s  g  Suppose  FCp(R)  •  c  v  of i d e a l s  > 2 .  tt  for a l l p e S  R  =  ©  K ^ ) ^ ^ (R) p  P  :  g  i t i s clear that  Rett  e  S)  * . g( ) f v  K  T h i s completes t h e p r o o f . Q.E.D.  -  -  100  Combining 4 . 3 - 1 and 4 - 3 . 2 we see t h a t i f .S i s anys e t o f primes then tt = © ('FC K-class f o r a l l cardinals  : p e S]  i s a generalized M  K > 2 . In f a c t . —  Now tt i s a l s o a K - c l a s s .  (  s = tt .  v  g(K)  3  I n the next theorem we s h a l l show  t h a t these a r e t h e o n l y c l a s s e s which a r e b o t h K - c l a s s e s generalized K-classes all  (except;  and  of course, f o r the c l a s s o f  rings).  4.3-3  THEOREM: Let  K  be a c a r d i n a l number  >_ 2  and l e t ft be a  c l a s s o f r i n g s which does n o t c o n t a i n a l l r i n g s . a strongly hereditary generalized K-class ft = © ( F C  p  : p e S]  Then ft i s  i f and o n l y i f  f o r some s e t o f prime numbers  S.  Proof: Assume t h a t  S  i s a s e t o f prime numbers and  ft = Q (FCp : p e S) . By 3 - 4 . 1 4 ft i s an e l e m e n t a r y r a d i c a l From 4 . 3 . 1  c l a s s so c e r t a i n l y ft i s s t r o n g l y h e r e d i t a r y . and  4 . 3 . 2 ft I s a g e n e r a l i z e d  K-class.  C o n v e r s e l y , assume t h a t ft i s a s t r o n g l y h e r e d i t a r y generalized K-class.  3" such  Then t h e r e i s a c l a s s of r i n g s  t h a t ft = ^ ( K ) * S i n c e ft i s s t r o n g l y h e r e d i t a r y ft <_ ft(2) (*  and by 2 . 4 . 2 ( v i i i ) , » < -B(2) = ( 3 g  (K)  g(K)  )(2) < ^  ft =ft(2)= ( ' ( K ) ) ( 2 ) 3  g  g  =  )(2) < ( * g ( K  g ( K )  )  < *  g ( 2 )  ) ) g ( 2 ) ^ *g(K)  ^g(2) '  N  o  ww  e s  h  a  1  =  R  1  '  g ( K  H  ) e  n  • *hus C  e  Prove t h a t i f  - 101 -  ft  FC  Suppose ft <t_ PC,  then ft i s the c l a s s o f a l l r i n g s .  then t h e r e i s a r i n g  Reft  such t h a t  PC(R) = (0) .  First  c " e ft .  we s h a l l prove t h a t  Let 0 4 x € R . Then <x> e f t so <x>/<x> e ft . 2 oo 2 <x>/<x> % C then nx e <x> f o r some i n t e g e r n . 2  If  Thus, t h e r e a r e i n t e g e r s nx = a^x Then  2  2  K + . ..- + a^x . L e t y = a^x  nx = yx  so  g e n e r a t e d by . Y = (<y>)  a , ..., a ^  2  such t h a t K 1 —  +  .. . +  a^x  <y> = n*Z = t h e i d e a l o f the i n t e g e r s  n . S i n c e ft = ft(2) , <y> e ft • Now c o n s i d e r  = the r i n g of  2x2 m a t r i c e s w i t h e n t r i e s from  <y> . Then e v e r y non-zero homomorphic Image o f Y  contains  a s u b r i n g g e n e r a t e d by one element which i s Isomorphic t o a homomorphic image o f Hence,  R  (2)  =  g  B  g ) > e ft(2) = ft . So i n any case  Since  c " e ft and j3 = t h e lower r a d i c a l {0°°} , j8 <.ft. Thus  ' C  03  e ft .  class  71= 0(2) <_ft(2)= ft .  Q(X) be the f i e l d o f r a t i o n a l f u n c t i o n s i n an i n d e t e r -  minate (Q(X)) Thus  Y e  C™ = <(°  determined by Let  <y> . T h e r e f o r e  X 2  over  Q = the f i e l d o f r a t i o n a l numbers.  i s a s i m p l e r i n g w i t h non-zero n i l p o t e n t  (Q(X))  e & ( )  2  g  2  =  R  '  B  u  t  t h e n  Then  elements.  , s i n c e ft =ft(2),  ( p [ X ] = < ( Q Q ) > e ft • Then a l l r i n g s g e n e r a t e d by one element a r e i n ft s i n c e ft s a t i s f i e s " c o n d i t i o n ( A ) . f o r e ft = ft / v = t h e c l a s s o f a l l r i n g s . g(2)  There-  0  S i n c e ft i s n o t t h e c l a s s o f a l l r i n g s , ft <_ PC .  - 102 -  Let  S = [p : ft / {(0)}} p  ft = ®  . Then, by Theorem 3-4.14,  (ftp : p e S} . We w i l l show t h a t  p o l y n o m i a l s i n an I n d e t e r m i n a t e  X  Fp[X] = the r i n g of  over  Fp  i s i n ftp and  hence ft = FC fora l l p e S . p p . Let (0) 4 <x> e ft . Then i f <x>/<x>  has a  2  ir  non-zero  n i l p o t e n t element,  of p-elements  Cp = t h e z e r o r i n g on t h e group  i s i n ftp . I f <x>/<x>  n i l p o t e n t elements  then  I n t h i s case  2  <x>  has no non-zero  has an i d e n t i t y so  Zp e ftp .  ( Z p ) e 'ftp <_ ft so s i n c e ft(2) = ft , Cp e ftp .  So i n any case' Cp e ftp . Now  (Fp[X])  2  e  = &  s  o  Fp[X] e ft(2) = & • T h e r e f o r e e v e r y r i n g o f c h a r a c t e r i s t i c p i s i n ftp . Now i f B i s . a n I d e a l o f a r i n g A and b o t h A/B  and B  a r e i n ft t h e n P  T h e r e f o r e , ft completes  p  A e ft ._ Hence 'P  = FC  p  f o r a l l primes  FC  < ft . P - P  p e S . This  the p r o o f . Q.E.D.  We now t u r n t o a c o n s i d e r a t i o n o f some c l a s s e s o f rings  W such t h a t  it = 7 ? n FC . g " g l  4.3-4  PROPOSITION: If  R e J  .  R e £> and  R  is finitely  g e n e r a t e d then  -  103 -  Proof: Let R e PC  be a f i n i t e l y g e n e r a t e d r i n g i n S)  R  then  Suppose t h a t R e ^ R  n  A R  so R e J  R e J3 fl PC  by 3-2.1.  R £ PC . L e t R = R/PC(R) .  <_ <£ , t h e r e i s a p o s i t i v e i n t e g e r  4 (0)  b u t R^  = (0) . Since  +1  n  such t h a t  R = Yl(A) . Now  i s f i n i t e l y g e n e r a t e d as a s u b r i n g o f R  Moreover,  W  1  0  has c h a r a c t e r i s t i c  p h i c t o a f i n i t e d i r e c t sum o f c o p i e s .of n  2 2  R  n  p ... P 2 R k  positive integers for  n  k  by 2 . 2 . 6 .  so from the Fundamental  Theorem f o r f i n i t e l y g e n e r a t e d a b e l i a n groups  2 R  Since  R e f ) , there i s a r i n g  w i t h D.C.C. on l e f t i d e a l s such t h a t n  . If  ? ... and s i n c e  R  i s isomor-  C™ . B u t then 2 R <3 A k  n  for a l l  t h i s c o n t r a d i c t s t h e D.C.C. c o n d i t i o n  A .  Thus we must have  R e FC  so  R e J  . Q.E.D.  Since J that  s  S l  -* Let  x e R . Therefore  8l  R  < D i t f o l l o w s immediately from 4 . 3 . 4  1  and  » - 4 • e  be a f i n i t e l y g e n e r a t e d r i n g i n <© and l e t  Now, R e J  = j6 n PC  <x> e jS fl PC = S  Notice = j>* and  so so  R  i s n i l p o t e n t and i n PC.  <x> eh-  I tfollows  that Proposition 4 . 3 - 4 also implies  fi'  = j>' .  that  that  - 104 -  4 . 3 - 5 THEOREM: .3" = 3" g g classes  =7|  "g  1  ,  fc,  £*  HFC i f  3" I s e q u a l t o any o f t h e  , J * , £)'  or  •  Proof: %  We have a l r e a d y n o t i c e d t h a t J  J<FC,  Since  J  Thus  < ^  g  R e || fl FC  zero homomorphic image o f such t h a t (and  <x>  <x> e FC J-*).  e  <-(J' )g  Thus  /  g  ( ;f*)  g  g  < (i * )  and  i  ^  a  g  and suppose t h a t  = J  < j3 = 72 g  g  R  i s a non-  R . Then t h e r e i s an  0 / x e R  and x  By 2 . 1 . 3 ( i i i ) (jf*)  / < jB , J  and s i n c e  = .A  ^ J*  i snilpotent. g  n FC < J < J'  (/*) < g  ( S')  so  &  /  g  <x> e f = f\  fl FC.  g  • Then by 2 . 4 . 2 ( i i ) , ( J ' )  < ( Cf')  &1  Thus  and by 2 . 4 . 2 ( i i i ) ,  g  • Notice  g  the above p a r a g r a p h we a c t u a l l y p r o v e d t h a t  Yl  g  that i n H FC <_ (  Now combining these i n c l u s i o n s we have Tig  n FC < ( / * )  1  npc<  and  < (j>*)  g i  (f*)  B' =  g  < ( ! %  J "  < (jf')  and  g  <(/')„•  Since  we need o n l y show t h a t  8*  ( f')  =  < Jl o  /* PI FC  o  t o complete t h e p r o o f . Since  j  5  <. FC , ( J') g  S  < Yi > J'  •  n PC- .  g  Let  < FC  g  f  .=  < 71  s  o  < (FC') = FC g  (f)cr < Tip.  •  Therefore  and s i n c e  - 1 0 5-  Q.E.D.  I n view of.-3.4.14 and. 4.3-2 t o determine tt f o r o  any e l e m e n t a r y r a d i c a l tt which i s to consider  3 where g b l o c k s o f 3-4.14.  4.3-6  3  <_ PC  i ti s sufficient  i s one of t h e b a s i c b u i l d i n g  PROPOSITION: = ((P })g  i f S 4 0 -  (1)  (3p(S))  (2)  ( 3 - T l ( S ) ) = F C n 72  (3)  (ar Yl(s))  g  p  p  g  p  p  if  g  = Pc n (& )  g  p  R  g  S = $ . if  s / pr .  Proof: Notice that since mentary c l a s s e s (  J  P  (l)  W  (  s  )  ( 3 (S.))  3p(S) and ^ 7 l ( S ) p  = (3" ( S ) )  are ele-  and  - Cp™ ", • 3  ) S  i  Suppose t h a t integers.  S i s a non-zero C.U.D. s e t o f p o s i t i v e  S i n c e e v e r y non-zero f i n i t e l y g e n e r a t e d r i n g  in  3" (S) c o n t a i n s a f i n i t e f i e l d o f c h a r a c t e r i s t i c p P i t i s c l e a r t h a t e v e r y non-zero homomorphic image o f a r i n g which i s i n potent  ( S ) ) c o n t a i n s a non-zero idemg such t h a t pe = 0 . Thus  e  P  (3 \ v( SJ)J) _< ({P p g}) . u  p  g  Jy  - 106 -  C o n v e r s e l y , suppose t h a t e v e r y non-zero homomorphic image o f R such t h a t set,  e 40  c o n t a i n s an idempotent S 4 P' and S  pe = 0 . S i n c e  i s a C.U.D.  <e> = Fp e 3" (S) so R e (3 (S)) . p p g T h i s completes t h e p r o o f . (2)  a: (0) = {(0)} , 3 71(0) - F C  Since (FC  (3)  1 e S . Hence  n 71 )  = FC  Suppose t h a t  Pi 71 and c l e a r l y  Jl .  f)  i s a non-zero C.U.D. s e t o f p o s i t i v e  Now ^ Y l ( S ) <  integers.  clearly  S  p  p  Z ^ y i { Z  +  = FC  )  p  n  i r  so  R  ( a T V L ( S ) ) <. P C fl ( £ ) . p  If  g  p  R  R e FC fl (dSr-p)  g  t h e n e v e r y non-zero homomor-  p h i c image c o n t a i n s a non-zero element x such t h a t p x = 0 N N-1 . f o r some i n t e g e r a >_ 1 and x = S a.x f o r some i n t e g e r s 1=1 a  1  a^,  a^._^  If x  i s not n i l then  commutative Wedderburn r i n g . y = y + (<x>)  <x>/7|(<x>)  is a  L e t y e <x> such t h a t  i s the i d e n t i t y of  <y>/Yl(<y>) = Fp e J ( S ) , thus  <x>/Yl(<x>) .  Then  <y> e 3" Y l ( S ) . So i n any  p  case, R e ( 3 : 7 2 ( s ) ) . p  Therefore,  g  (3T y i ( S ) ) p  g  = FC  p  n (£  R  )  g  . Q.E.D.  - 107  k.h  -  THE GENERALIZED RADICAL CLASS . A ring  R e £  £  g i  i f and o n l y i f R  R e £  i f and o n l y i f f o r each x e R N u. , v. e R such t h a t x = E u.v. . 1 1 i i  2  = R .  Hence  t h e r e a r e elements Using t h i s  characteri-  i = 1  z a t i o n of  £  one e a s i l y sees t h a t  Let  R e £  g  .  Since  £  R/R  2  i s a radical  class.  cannot c o n t a i n a non-  2  . zero idempotent s u b r i n g , R/R = (0) . Thus 2.4.2(iii), £ < £ so £ < £ < £ . g ~ g g]_ g -  R e £ .  By  v  x  Let  R  be the s u b r i n g o f the r i n g o f r e a l numbers  g e n e r a t e d by t h e s e t o f p o s i t i v e . r e a l numbers {+(2) l/2  2 ' R  : n > 1}  1 / / 2  n  .  then  .  Clearly /  I  2  = R  since  [(2)  1 / / 2  ]  2  = .  However, i f R  7  R  T h i s f o l l o w s from a lemma on page 215  / (R )  7  .  7  i s a f i n i t e l y generated subring of  of Z a r i s k i and Samuel [15] and  R  which i m p l i e s t h a t i f  (0) / I = I  i s f i n i t e l y g e n e r a t e d as an i d e a l of a commutative  r i n g then  I  has an i d e n t i t y .  Thus  £ ^ £ . The r e s u l t from Z a r i s k i and Samuel . g implies that £ = £ f o r commutative r i n g s b u t we do not g^ g know i f £ (R) = £ (R) f o r a l l r i n g s R . l . s  s  Notice that  R e £  i f and o n l y i f e v e r y non-zero l contains- an idempotent e / 0 , and s  homomorphic image of that potent  R  is  £  s  e / 0 .  l  R  s e m i - s i m p l e i f and o n l y i f R  has no idem-  2  - 108  e' < e  Clearly  (£'.) = (£') g e  R  )' = (e ) ' = e ' l  s  < FC n £ 1  i n t e g e r s moduls 4 i s n o t i n If  (e  so l  s  course,  -  s  x  (£')  .  Since the r i n g of  , (£')  £ FC fl £  R  .  (0) / I  i s a commutative r i n g and  f i n i t e l y generated i d e a l of  . Of  s  which i s i n £ s  i sa  <_ £  then  l  p  I  = I  of & g  R . l  so  I  has an i d e n t i t y . . Thus  I  i s a d i r e c t summand  T h e r e f o r e , f o r a commutative r i n g  R  w i t h A.C.C.  (R) has an I d e n t i t y and I s a d i r e c t summand o f  we do n o t i n s i s t t h a t  R  R . If  be commutative we o b t a i n t h e  f o l l o w i n g weaker r e s u l t .  4.4.1 THEOREM: If  (OW  R e &  and  R  has A.C.C. on o n e - s i d e d 2  i d e a l s t h e n t h e r e i s an R = ReR  and  e  e e R  such t h a t  i s an I d e n t i t y f o r  = e / 0 ,  e  R/Yl(R)  4  (0)  .  Proof: Let  (0) / R e £ g  on l e f t i d e a l s .  Then  so by G o l d i e ' s Theorem a l e f t quotient r i n g  and suppose t h a t  R  has A.C.C.  l  ( 0 ) / R = R/7^(R)  i s semi-prime and  (Theorem 29 i n D i v i n s k y [ 7 ] )  R  has  Q, which i s a f i n i t e d i r e c t sum o f  matrix r i n g s over d i v i s i o n r i n g s .  C l e a r l y a l l idempotents  of  Q  are i n the  are i n the  centre Since  R  . • Choose  t e n t of  centre  .  f e centre  R/Rf  .  i s an idempotent and Rf  w e R  One  R  i s an idempotent  i s n i l and R = ReR  .  Is an  R  w  Rf c Rf'  .  idempoIf  Rf = R  Rf / R  i s a non-zero f ' = f +w ,  By  This implies that  such t h a t  Rf = R , ReR  because  such that  Therefore,  hence does not  : e  R , Rf <3 R  Then by Lemma 1.12 e e R  Since  of  ff' = f ,  , Rf = Rf'  .  [Re  e a s i l y checks that  since  which i s a c o n t r a d i c t i o n . identity for  of  there i s a non-zero Idempotent i n  Sl  then there i s an element  maximality of  so a l l iderapotents  maximal i n the s e t Since  idempotent i n  Q  R .  R e £  Rf  R)  of  of  so  i n Herstein  - fw  the  w e Rf f  i s an  [10]  there  e + Yl(R) = f .  + N = R  .  Therefore  R/ReR  c o n t a i n a non-zero idempotent.  Thus  R e £ Sl  The we  proof  assume t h a t  R  goes through i n e x a c t l y the same way has  A.C.C. on r i g h t  if  ideals. Q.E.D.  The  following proposition  2.4.13) i m p l i e s that i f and  M  contains  all  £  3_  i s an elementary semi-simple c l a s s semi-simple r i n g s then  s  c l a s s of  M  (together w i t h Theorem  semi-simple r i n g s f o r some c l a s s  s.i s a t i s f i e s condition  M =  the  l  (A).  5"  which  - 110  4.4.2  -  PROPOSITION: 34 i s an elementary c l a s s of r i n g s which  If tains a l l £ s  34 s a t i s f i e s  semi-simple r i n g s then  con-  condi-  l  t i o n s ( 2 ) i f and only i f 34 s a t i s f i e s  condition  s"(2).  Proof: Let  34 be an elementary c l a s s of r i n g s which  tains a l l £ s  If  con-  s.s. rings. l 34 s a t i s f i e s  c o n d i t i o n s"(2) then by 2 . 4 . 1 2  34  s a t i s f i e s condition s ( 2 ) . Suppose 2.4.9  34 s a t i s f i e s c o n d i t i o n s ( 2 ) .  34 i s a semi-simple c l a s s .  JJ(<X>) = I 4 ( 0 ) • ^ < £  .  Since  Therefore  Therefore <x>  and  ring i n  I I  J  I e £  of  <x>  <x>  34 then  34 c o n t a i n s a l l £  g  s.s. rings,  so by v i r t u e o f the remarks  a t the b e g i n n i n g of t h i s s e c t i o n there i s an i d e a l  If  By Theorem  I  has an i d e n t i t y and so  such that  I © J ' = <x> .  i s generated by one element as a s u b r i n g of cannot be homomorphically mapped onto a non-zero  34 . We have proven the c o n t r a p o s i t i v e of c o n d i t i o n s ( 2 ) .  This  completes the p r o o f . Q.E.D.  In the f o l l o w i n g p r o p o s i t i o n we show that i f Ji i s  - Ill  -  an e l e m e n t a r y s e m i - s i m p l e c l a s s which I s n o t o f t h e type w i t h which 4 . 4 . 2 i s concerned then ti must be c o n t a i n e d i n the  c l a s s o f *)2  ^0-  n  S  4.4.3  tion  s  p.  PROPOSITION: 3  If  r  s e m i - s i m p l e r i n g s f o r some prime P  r ( 2 ) then  be a' c l a s s o f r i n g s which s a t i s f i e s 3  >1  >_ "ft H FC •' l S  f o r some prime  condi-  p  or  P  < e  i  s  i  Proof: 2T be a c l a s s o f r i n g s which s a t i s f i e s  Let  condi-  tion r(2) . Suppose t h a t f o r a l l primes Then f o r each prime which i s  3" s  l  p  there i s a r i n g  s.s.. Since  3  p ,  3_  £ Tin- H PC . g P  Rp e Yl fi PC  s a t i s f i e s c o n d i t i o n r ( 2 ) , by  Theorem 2 . 4 . 7 , t h e c l a s s o f  3" s . s . r i n g s i s an e l e m e n t a r y l T h e r e f o r e , f o r each prime p t h e r e i s a r i n g s  class.  <x > c Rp p - ^  such t h a t  Since subdirect C  00  is  <x > p  i s 3" s . s . and g x  <x > = Cp . P F  sums o f s e m i - s i m p l e r i n g s a r e s e m i - s i m p l e ,  3  s . s . Now e v e r y z e r o r i n g on a c y c l i c group i s l 3 s . s . so s i n c e the c l a s s o f 3' s . s . r i n g s i s e l e m e n t a r y l l s  g  s  and s a t i s f i e s c o n d i t i o n If  R  £ s  l  (P) a l l n i l r i n g s a r e  3^ s . s .  t h e n t h e r e i s a non-zero homomorphic  -  image in  R  R .  of Let  R  -  112  such t h a t t h e r e a r e no non-zero idempotents  0 / x e R  (0) / I <I <x> .  and  a r e no non-zero idempotents i n  <x>  , 1 / 1  be homomorphically mapped onto the non-zero  .  Since there Thus  I  can  3" s . s . r i n g l s  2 I/I  .  Therefore  <x>  is  3" s . s . S i n c e the c l a s s o f l 3 s . s . r i n g s i s e l e m e n t a r y , R i s 3 s . s . so R i J g-j_ g-j_ l s  s  Therefore  3 s  l  <_ S s  l  .  T h i s completes the p r o o f . Q.E.D.  - 113 -  The r e l a t i o n s h i p s between the g e n e r a l i z e d  radical  c l a s s e s i^hich we have d i s c u s s e d can.be i l l u s t r a t e d I n the following  diagrams.  fC  The f o l l o w i n g diagram remains v a l i d when i s everywhere r e p l a c e d  by  PC  FC 1?  '3,  £  ILLUSTRATION 2  9j  f\  FC  - 114- -  CHAPTER V LOCAL RADICAL CLASSES  5.1 THE RADICAL CLASSES  ?.&.*  The c l a s s  AND  .*  £ .  i s a l o c a l r a d i c a l c l a s s by  Theorem 2 . 2 - 7 -  If  Re £  R' = <x^, ..., x >  and  N  R' = (Sa.d. : d. e D ^  1  1  D = (x. l  1  ... x. L  x  R e 3 . © . *  then  R " = (0)  Since that  a.  : L <_ K - 1} .  are Integers}  1  where  However, f i n i t e f i e l d s a r e  1  3 . ft.* b u t n o t i n £ .  in  then  /K  and the  for i f  £ ^ 7\  3 . & . * i 71  .  3 . 8 .* ^ £ .  Therefore,  the f o l l o w i n g p r o p o s i t i o n i m p l i e s 3.9.*  Hence  i s unrelated  to %  .  5.1.1 PROPOSITION:  z =  .* n Yl .  3.S  Proof: Since £  <_ 3 .  & .* D  £ <_ 3.B  .*  and  £ <_ 71  "R .  Assume  R e 3.Sb .* n VI  generated s u b r i n g o f  R .  •  Then s i n c e  s a t i s f i e s A.C.C. on o n e - s i d e d I d e a l s • group  R  / +  i t i s clear that  has A.C.C. on s u b g r o u p s ) .  L e t R' R  7  be a f i n i t e l y  e 3 . & .* , R '  ( i n f a c t , the a d d i t i v e Now  R' e y) and  - 115  -  = j3 f o r r i n g s which s a t i s f y A.C.C. (Theorem 16 i n D i v i n sky [ 7 ] ) so  R' e |3 <_ £ . Thus  R'  i s n i l p o t e n t so  R e £.  Q.E.D.  Since  |3 <_ £ , £ * < _ £ * = £  and s i n c e a l l  r i n g s a r e i n 0 , £ <_ J3* . T h e r e f o r e ,  nilpotent  £ = £* = |3* .  5 . 1 . 2 THEOREM: If then a r i n g  34* i s a l o c a l r a d i c a l c l a s s and R  /3 <_ 34* <_  i s 34* s e m i - s i m p l e i f and o n l y i f R i s  i s o m o r p h i c t o a s u b d i r e c t sum of prime  34* s e m i - s i m p l e  rings.  Proof; Let  0  34* be a l o c a l r a d i c a l c l a s s such t h a t  ^ 34* <_ 3. & . * . S i n c e s u b d i r e c t sums o f s e m i - s i m p l e r i n g s a r e s e m i -  s i m p l e one d i r e c t i o n i s c l e a r . C o n v e r s e l y , suppose t h a t  R  i s 34* s e m i - s i m p l e .  I t i s s u f f i c i e n t t o f i n d , f o r each non-zero l(x)  such t h a t  x \ l ( x ) and  R/l(x)  x e R  i s a prime  an i d e a l 34*  semi-  simple r i n g . Let  0 4 x e R .  f i n i t e l y generated s u b r i n g Let  Since  ( x ) \ 34* t h e r e i s a  R' C ( X )  R  such t h a t  r  Z ( x ) = ( I < T R : R'/R' fl I | 3i*} .  an a s c e n d i n g c h a i n i n Z ( x ) and l e t  Let  J  J = U(J  Q  R' : a e A  34* . be  : a e A] . I f  -  n J e 3.  a.  Now by  J  R ' - fl J = R ' fl J a  choose  .  2  .  6  .n J  R'  -  . R/I(x)  F i r s t we s h a l l prove t h a t Let K O R  R'/R'  Now,  R  i(x)  R'  n  R' n  e  K  **  _  R  ,  T  H  n  K p l(x) .  such t h a t  n  x  E  n i(x)  R'  N  €  M  R^HTH  •(R  J  R' n K „ R' R' n H n K =  6  RV(R' n n K)/(R'  S  O  R  |  R  n i(x)  i(x)) n i(x))  „ =  R  J  This i s a c o n t r a d i c t i o n .  6  e  r  R' n K  f  o  r  «*  h  u  s  l  f  e  •  +.  i  34* i s s t r o n g l y  R/I(x)  i s 34* s e m i - s i m p l e  R/I(x)  i s prime we b e g i n R  a r e i d e a l s of  such t h a t  K fl H = l ( x ) . Then t h e r i n g  n K + R' n R' n H  w  e  T  K 0 H p l ( x ) . Suppose t h i s  then  i s n o t t r u e ; t h a t i s , suppose  =  h  since  and H  H p l(x)  and  T  ,  K  by showing t h a t i f K  R'  '  *  I n o r d e r t o prove t h a t  K p I(x)  '  R  K / l ( x ) i 34* . Hence  hereditary,  n K n i(x)  Then  = ( ' n K ) / ( R n i(x))  K  + i(x) i l(x) — *  R'  pri—p j ( ) =  R'  i s 34* s e m i -  e 34* .  K  fl  a e A .  f o r some  a  J e Z ( x ) so by Zorn's Lemma we may  I ( x ) maximal i n Z ( x )  simple.  i s finitely  which I s a c o n t r a d i c t i o n s i n c e  e z(x) . Therefore  a  2  R' fl J c R ' n J  g e n e r a t e d as a r i n g so Thus  -  R ' / R ' fl J e 34* < 3.ft.* so  J £ Z ( x ) then R'/R'  116  *  •  E  *  N  V  *  o  H  Therefore  3  . a  s  w  b  r  .  i  n  »  s  o  **  '  w  . S  . 1  O  R'  R' n i(x)  £  K fl I p l ( x ) .  f  - 117  -  Now we can.prove t h a t Suppose t h a t Then  K  and  K fl H ^ l ( x )  /3(R/I(X))  4  (°) •  34* s e m i - s i m p l e and Since This  H  R/I(x)  i s a prime r i n g .  a r e as above and t h a t (K fl H )  and  2  c l(x) .  KH c l ( x ) .  Then  This i s a c o n t r a d i c t i o n since jS <_ 34* .  Therefore  R/l(x)  0 I ( x ) k 34* , R ' £ l ( x )  R'/R'  R/l(x)  is  I s prime. so  x § I(x).  completes the p r o o f . Q.E.D.  I n view o f 1 . 1 . 6 l o c a l r a d i c a l classes  t h i s theorem i m p l i e s t h a t a l l  34*  w h i c h a r e between  are s p e c i a l r a d i c a l c l a s s e s . special radical class.  Divinsky  and  In particular,  Theorem  p r o o f f o r Theorem 52 (£  0  .*  $.&.* is a  p r o v i d e s an a l t e r n a t e  i s a special radical class) i n  [7].  We conclude t h i s s e c t i o n by c o n s i d e r i n g the g e n e r a l i z e d and e l e m e n t a r y c l a s s e s r e l a t e d t o By 2 . 4 . 2 ( v i i )  £  = £ g  (3.B.*)  and  £  S  £ = ]/} and g S (Theorems 4 . 2 . 2 and 3 ' 3 « 4 r e s p e c t i v e l y ) . Since a r i n g  <x> e 3 . f i .  are i n t e g e r s  a^,  x  + ... + a,x = 0  n  + a_ , x  n _ 1  a n  _ ]  L  2 . £).*. s  =(3.S.*)  g-^  We have a l r e a d y seen t h a t  and  j  g-^ <£' = VI  i f and o n l y i f t h e r e  such t h a t i t i s clear that  - 118 -  (3.fc.*)  = ( &  g  R  )  g  and  (ff.S . * ) ' = &  5.2 THE LOCAL RADICAL CLASSES Let  *, £) *  R  AND  =  •  Fl* .  F l be the c l a s s o f a l l f i n i t e r i n g s .  s h a l l begin t h i s section  w i t h a d i s c u s s i o n o f those r i n g s  such t h a t e v e r y f i n i t e l y g e n e r a t e d s u b r i n g o f I n the f o l l o w i n g  We  proposition  R  R  i s finite.  we c o l l e c t s e v e r a l e l e m e n t a r y  p r o p e r t i e s of t h i s c l a s s of r i n g s .  5.2.1 PROPOSITION: (1)  F l * Is a radical £ F l * £ 3 . & .*  (2)  class. .  (3)  £' £ F l * £ FC .  (4)  FI* = 3.  £>.* n FC  (5)  £) and  F I * are unrelated.  (6)  FF  and  .  F I * are unrelated.  Proof: (1)  Clearly so  (2)  the c o n d i t i o n s of Theorem 2.2.7 a r e s a t i s f i e d  FI* i s a local radical <J = j3 fl FC  Since  class.  any f i n i t e l y g e n e r a t e d  3  r i n g must  be f i n i t e and o f course any f i n i t e r i n g i s i n t F . D . Therefore, J  < F I <_ 3 . & .* .  S i n c e the r i n g o f i n -  t e g e r s i s i n 3.S> .* , F I * £ 5 . £ .* . but  F  <t  J  so <f £ F I .  The r i n g  F  p  e FI*  - 119  -  (3) . By 3 . 4 . 4 £' <_ F l and s i n c e any f i n i t e n i l p o t e n t r i n g i s i n F l * , t' (4)  £ F l . Clearly  P I * £ FC .  T h i s i s c l e a r s i n c e a r i n g i n 3. £ . fl PC  must be  finite. (5)  Since  F  e PI* , PI* £  p  . Example 6 i n R i n g s and  R a d i c a l s D i v i n s k y [ 7 ] shoxtfs t h a t t h e z e r o r i n g on the • additive £) i  (6)  group o f r a t i o n a l numbers i s i n £) so  PI* .  S i n c e f i n i t e f i e l d s a r e i n P I * , F l * £ FF . Q < FF  FF £ P I * s i n c e  and  Clearly  £) £ F l * . Q.E.D.  The f o l l o w i n g characterization  theorem p r o v i d e s an i n t e r e s t i n g  of P I * .  5 . 2 . 2 THEOREM: R e P I * i f and o n l y i f every f i n i t e l y subring of R  s a t i s f i e s D.C.C. on l e f t  generated  ideals.  Proof: S i n c e a l l f i n i t e r i n g s s a t i s f y D.C.C. one d i r e c t i o n Is  clear. Conversely,  subring of the r i n g Let  assume t h a t e v e r y f i n i t e l y R  s a t i s f i e s D.C.C. on l e f t  generated ideals.  R ' be a f i n i t e l y g e n e r a t e d s u b r i n g o f R and  - 120 -  N' = 7 l ( R / )  let  • The r i n g  R'/N'  i s Isomorphic t o a  f i n i t e d i r e c t sum o f m a t r i x r i n g s over d i v i s i o n r i n g s . R'/N' ~ (D ) Q •• . O (A ) ' • Then each 1 n^ k n  g e n e r a t e d as a r i n g and i f A^ r i n g of Then  D.  <x^>  D.  w  then  A.  L e t x. e D. .  f o r some i n t e g e r  n+  i s finitely  1  i s a f i n i t e l y g e n e r a t e d sub-  s a t i s f i e s D.C.C.  = <x^-> ^  n  n >. 1  L f«o r some i.n , x.n = a ^ - x n+1 . + ... + a.x. tegers l n+1 l i i ° Since  D^  x  n + 1  ±  =.a  3.4.4,  has no p r o p e r d i v i s o r s x  D  by 2 . 2 . 6  + ... + a x ^ "  2  n + 1  L  .  i sa finite field.  ±  N'  By 3 . 2 . 1  J  Thus  e e'  D  ±  R'/N'  I  N' e J*  = j3 n PC  so  s  R'  T  so by  f i n i t e and so Since  N'  by Lemma 28 i n N'  i s finitely  g e n e r a t e d , n i l p o t e n t and of f i n i t e c h a r a c t e r i s t i c . i s f i n i t e so  .... a . ' L  n+1'  of zero  Therefore  s a t i s f i e s D.C.C.on l e f t i d e a l s  so a  i s f i n i t e l y generated as a r i n g .  Divinsky [7].  Let  Thus,  N'  must be f i n i t e .  Therefore  R e PI* . Q.E.D.  We now t u r n t o a c o n s i d e r a t i o n o f the l o c a l c l a s s e s jf *  5.2.3  and  g>*  . By 4 . 3 . 4  j>* =  £* .  PROPOSITION: j *  = £ n FC  =  z n PI*  =  y>  nFI*  - 121  -  Proof: J  Since R e £ fl PC  < PC  J  and  < X , J *  < ^ n PC  then e v e r y f i n i t e l y g e n e r a t e d  .  If R  s u b r i n g of  is  n i l p o t e n t and of f i n i t e c h a r a c t e r i s t i c and hence i s i n jB fl PC = J  ; thus Since  R e J  * .  Therefore,  , £ P. P I *  P I * < PC  J * = £ 0 PC .  < £ fl PC = J *  .  Now  a  f i n i t e l y generated n i l p o t e n t r i n g of f i n i t e c h a r a c t e r i s t i c J * < Fl* .  i s f i n i t e so  As above  J*  <  r i n g i s n i l p o t e n t and i n  J>* = £ n F l * .  Thus y^fl  PC  Fl* .  ,  Since a f i n i t e n i l  "ft fl P I * <_  J** .  T h i s completes the p r o o f Q.E.D.  The f o l l o w i n g i s a s l i g h t m o d i f i c a t i o n of an example g i v e n by Baer [ 6 ] . that  £ fl FC £ 0  5.2.4  EXAMPLE:  We p r e s e n t t h i s example t o show  and hence t h a t  F o r each i n t e g e r a d d i t i v e group of 2  k  J*  £ jB .  let  = ( 0 , a ( k ) } =.the  elements. CO  F o r each i n t e go e r such  i _> 1  let  CO  T. e HomfS G, k' . — 00 v  x  that , 0 i f k s 0 mod 2 . T(a(k))o{ a ( k - l ) i f k £ 0 mod 2 1  x  1  .  —CO  E Gk' ,)  - 122 CO  Let  CO  be the s u b r i n g o f Hom( S G , £ G )  R  k  i s g e n e r a t e d by the s e t (1)  (0) 4 I <! R  be any i n t e g e r Now  X  may w r i t e V, 4 0 K  Let  and l e t 0 4 X e I . L e t h  i s a sum of monomials i n the + .... +  and V.  V, = T k m , - 1,0 . V  2 I  k  where  h  / (0) .  T^'s so we  I <_ k ,  4 0 ,  i s a sum o f monomials o f l e n g t h  1  i+1 .  ... T +...+T ...T m, mn,0 ~ mn,k , l,k  4 0  V (a(t)) 4 0 .  t h e r e i s an i n t e g e r  (We may choose  k  mod 2  /3(R) = (0) .  >_ 1 . We s h a l l prove t h a t  X =  Since  -09  (T\ : i >_ 1} .  F i r s t we s h a l l prove t h a t Let  which  fc  -OS  t > 0  t  such t h a t  since  t =0  i f and o n l y i f - t s 0 mod 2 ) .  1  1  Choose i n t e g e r s  r  and  m. .'_<_ r  and  0 < j < k , and 2  and s  for a l l i r + 1  and j -2  Consider  2  r (T " 2  k - 1  r  k  h  1 <_ i <_ n  .  s  .X)  r  2  h  (a(t + 2 -2 ))^ 0 , h  r  s  V ( a ( t + cp-2 )_)  • such t h a t 1 < cp <_ 2 . * k- " mod 2 f o r some i and 0 <_ j <_ k . S i n c e m. , . mod 2 ' " J .. Thus t -  k _ 1  k+1 < 2  such t h a t  + t < 2  h  r (T ~  We w i s h t o show t h a t To b e g i n we c o n s i d e r  such t h a t  V,') ( a ( t + 2 - 2 ) ) . h  where  Suppose  r  cp i s an i n t e g e r  t + cp2  - j =0  r  m  ±  1  }  J  K  implies that  and j such t h a t 1 <_ i <_ n r >_ m. , . , cp2 = 0 . i ^ - J _ _ X K j & 0 mod 2 > ~2 . T h i s  T ... T m. , i,^ 0 im. ,k  r  m  maps ^  a ( t + cp.2 ) ^ r  v  1  onto  - 123 -  0  i f and o n l y i f i t maps  V^(a(t)) 4 0 a(t)  onto  a(t)  onto  0 .  Since  an odd number of the monomials must map a(t-k-l)  V ( a ( t 4-cp-2 )) =  so  r  k  a(t+cp-2 -k-l). r  T 2^—k—1 ( a ( t + cp-2 r - k - 1 ) ) .  L e t us now c o n s i d e r Suppose Now  i s an i n t e g e r such t h a t  t + cp-2 < 2 r  <_ 2  1  I  t+cp-2  r  r  k + 1 + I < 2  -2  r  ~  ( a ( t + cp.2  T  2  " " ( a ( t + cp-2 1  r  - k - 1)) 4 0  r  - k - 1))  = a ( t + cp«2  - k-  r  .  Since  = t + (cp - l ) 2 > 0.  r  r  r  2  s  - k - 1.  r  so  r  0 < t + c p - 2 - k - l - - f , <2  T  k _ 1  1- <_ I <_ 2  t + cp-2 - k - 1 < 2  - k - 1 - I >_ t + cp-2  Therefore,  k  so  s  k - 1 ,  r  s  so  S  so  l - ( 2  r  - k  - 1})  = a ( t + (cp - l ) 2 ) . r  I t follows that 2 -k-l  2  r  ( T  Now i f I = k 0  If where  4  ^ f '  k  then  ~ \ f 2  ( a ( t  i  monomial a p p e a r i n g i n  2  h  m  2  v  )  )  =  Z  (  t  ^  )  0  •  = Z + ( T ~ ~ ^ • V, ) s k k  v  i s either > t .  a  so  2 r  7  r  y  +  X =  2  Z(a(t + 2 .2 )) h  h  e I ^  h  £ < k , (T *.X) * ' s  a ( v ) ' s where each i  # V k )  0  2  h  y  or a sum of  ( S i n c e the l e n g t h of z  i s s t r i c t l y l e s s than  - 124 -  ( ( k + 1) + 2  r  a(t + 2 -2 ) h  r  - k - 1)2 = 2 2 as " f a r down" as  Therefore 2 case  t h e y cannot "move"  r  0 / (T ~ 2  k _ 1  -X)  a(t)). e l  2  I  4 (0) .  T h e r e f o r e no non-zero  m = max{s : T  o c c u r s i n some  Z e (R')  length a t least one o f  (3)  h  k , k-1,  k-h  Z(a(k)) = 0  Thus  (R')  Therefore  Since  m  , 2  h > 2m  Choose  i s of  must d i v i d e  m  f o r a l l integers  f o r a l l integers  k  so  k . Z = 0 .  R e £ fl FC ( s i n c e  Therefore  2R = ( 0 ) ) Since  so  R  i s j8  £ fl PC = J * ,  * £ /3  From P r o p o s i t i o n 5.2.1 (4) we know t h a t J * <_ P I * ,  so s i n c e 5.2.4 J* P i  we see t h a t and  J  *  There-  = (0) .  h  s e m i - s i m p l e and i n £ fl FC . R e $ * .  R . Let  X.} .  h > 2  R  1  then each monomial i n Z  h .  fore  i d e a l of  L e t R' = <X ,...,X >  be a f i n i t e l y generated s u b r i n g o f  Now i f  So i n any  h  i s n i l p o t e n t so j3(R) = (0) . We s h a l l now prove t h a t R e £ .  (2)  .  2  £ J * .  and J * *  Combining t h i s  are u n r e l a t e d . -  £ FI* with  The c l a s s e s  j8 a r e a l s o u n r e l a t e d s i n c e i t i s c l e a r t h a t •  The r e l a t i o n s between these r a d i c a l s c l a s s e s can be i l l u s t r a t e d by t h e f o l l o w i n g diagram.  - 125  -  ILLUSTRATION 3  -  Let  -  J**  be a non-zero  on l e f t i d e a l s .  D.C.C.  sum  R  126  Then  semi-simple r i n g w i t h  R/y)(R)  is a finite  of m a t r i x r i n g s over d i v i s i o n r i n g s . on l e f t I d e a l s  D.C.C.  fl  7\{R)  y|(R) .  £ ( R )=  Since  direct  R  satisfies  Thus  = £(R) n F C ( R ) = J p * ( R ) = (0) so F C ( R ' ) i s  FC(R)  i s o m o r p h i c t o an i d e a l o f R / Y l ( R )  and hence i s a f i n i t e  d i r e c t sum o f m a t r i x r i n g s over d i v i s i o n r i n g s o f f i n i t e characteristic.  0 / x e FC(R/y2(R)) an  identity  e  and  e e FC(R/7f(R))  so  k , this implies that e = e  k  FC(R/YI(R)) = ( F C ( R )  R  e Yl  t h e n  completes  5.2.5  e  (ne) k  k  J R  e  <_  J*  = 0  Therefore  contains  ne = 0 f o r  f o r some p o s i t i v e  integer  . Therefore,  + yi(R))/n(R)  •  so by Lemma 28 i n D i v i n s k y [ 7 ] i f  *  -  .  x  e F C ( R ) so  e ( F C ( R ) + 77(R))/y?(R)  Now  such t h a t  , then t h e i d e a l g e n e r a t e d by  n / 0  some i n t e g e r  x e R  Suppose t h a t  Since  R <t J> * , ")/|(R) / R  .  This  t h e p r o o f o f t h e f o l l o w i n g theorem.  THEOREM: If  (0) 4 R  i s a J>*  D.C.C. on l e f t i d e a l s then  semi-simple r i n g w i t h  R k Yl , FC(R)  is a finite  d i r e c t sum o f m a t r i x r i n g s over d i v i s i o n r i n g s o f f i n i t e c h a r a c t e r i s t i c and R/(FC(R) + YJ(R))  i s a f i n i t e d i r e c t sum  of m a t r i x r i n g s over d i v i s i o n r i n g s o f c h a r a c t e r i s t i c Notice that i f R  i s not only  0 .  semi-simple  - 127  but a l s o  -  P I * s e m i - s i m p l e then a l l the d i v i s i o n r i n g s a r e  infinite.  If  summand o f  e PC  (R/7J(R))  R ; i n fact,  then  i s a direct  J}(R)  R = Y|(R) © P C ( R ) .  Prom Theorem 13 i n D i v i n s k y [ 7 ] we conclude ^*(R)  may n o t be e q u a l t o  However, i f R e PC  since  Yl(B.)  R e ^f*  1 ' = Tin  R  i s a non-zero r i n g w i t h A.C.C.  /3(R) = £ ( R ) = 7 i ( R )  Then  j f ( R ) = J * ( R ) = $'{R) and  Ot c o u r s e , as we  i f and o n l y i f R e ) |  Suppose now t h a t on l e f t i d e a l s .  since  >  J * ( R ) = X ( R ) n FC(R) = £ ( R ) = 71(H) • n o t i c e d above,  £ $* .  J *(R) = 7l(R)  then  that  so  J = /3 fl PC , J** = <£ n PC  since  PC .  U n f o r t u n a t e l y we cannot use G o l d i e ' s Theorem t o obtain a r e s u l t s i m i l a r to 5.2.5. <§ * s e m i - s i m p l e , C  m  R  consider the r i n g and R  may be i n 0 ( f o r example,  i s j f * semi-simple).  not be the same as  Even i f R $ /3 ,  ( F C ( R ) + /3(R))//3(R) . R = (p[X]/(2X )  s a t i s f i e s A.C.C. b u t  R  .  <x>  Clearly  C  ro  e j3 b u t  FC(R/j8(R))  may  To see t h i s Then  FC(R) = (X ). 2  F C ( R ) D Y|(R) = (0) and  R / y f ( R ) e PC .  However, i f R e PC  <x> e %• 0 PC  .  2  YXR) = J8(R) = (2X)  A ring  F i r s t of a l l , I f R i s  then c l e a r l y  j3 ( R ) = J*(R)  i s f i n i t e I f and o n l y i f  so i t f o l l o w s t h a t  (PI*)' = L  n FC  and  .  - 128  = (FI*)  (PI*) g v  y  = (&„)  c l e a r the  n FC .  J * = £ n PC  Since  ;  (J*)' =  7}n PC  and  (J*)  =  v  H FC .  a  From 2 . 1 . 3 J" <  So- < g  p c  i t is  R g  'gjL  v  -  s o  o f  (iii), course  < .J ' J**  o  and c l e a r l y  = £ n ^'  = £ n «f  v  The r e l a t i o n s h i p s  . s  between these r a d i c a l c l a s s e s  can be i l l u s t r a t e d by the f o l l o w i n g  diagram.  - 129  -  Z 4 tt 4 71 .  5-3 LOCAL RADICAL CLASSES tt FOR WHICH  6* = <£* = Z .  •We have a l r e a d y seen t h a t  F o r any  c l a s s tt , tt* <_ tt' so from Theorem 3 . 3 - 2 we may conclude t h a t tt* = y?  f o r a l l classes of r i n g s  YJ <_ tt <_ FF .  Since  Yl < Yl  and  tt  FF  s  <_ FF  of r i n g s  &  e>^  e>  tt  R e c a l l that  f o r a l l classes  &2_  7? 1  such t h a t  1  M  i t follows  s  ( t t ) * = (tt )* = (tt )' = (tt )' =7?  that  such t h a t  F  F  •  1£ i s t h e upper r a d i c a l c l a s s  mined by the c l a s s o f a l l s i m p l e idempotent r i n g s . 0 <U<  s  F  VJ  o  = YL -  t l * = YL  prove t h a t  yj*  Therefore  <_  deterClearly  = YL  -  To  i t i s s u f f i c i e n t t o show t h a t a f i n i t e -  l y g e n e r a t e d n i l r i n g cannot be homomorphically mapped onto a s i m p l e idempotent. r i n g .  I n f a c t we can prove the f o l l o w i n g .  5 . 3 - 1 PROPOSITION: A non-zero f i n i t e l y g e n e r a t e d n i l r i n g i s n o t idempotent.  Proof: Let  (0) 4 R = < x , .•. ., x > 1  n  be a f i n i t e l y  generated - n i l r i n g . 2  Suppose t h a t a minimal subset  R = R  fx.. , l  R = Rx. 1  1  1  + ... + Rx. k x  . Choose  r  x. } k  x  that  == Rx, + ... + R of  (x,, i  n  x } n  such  - 130 -  Since  x^  e R  t h e r e a r e elements  \> ' ' ' > \t  v  v  e  R  such t h a t x. x  Since for  R  = r x . +-...+ r, x. l k  (*)  n  l  1  X  K  i s n i l r, -x. m  some i n t e g e r  '= O ' X . '  m >_ 1 . L e t  1  = 0 e Rx. t  + ... + Rx.  be the s m a l l e s t  integer  P  >_ 1  such t h a t  x, x. 1 1  •t—1  r, 1  x.  Since  f  = r"x. 1 i  1  f  1  e Rx. 2 1  -1  + r,' r„x. 1 2 i  -t i s m i n i m a l ,  + ... + Rx. k  .  Then from ( * ) ,  x  •£ —1  2  + ... + r,' r, x. 1 k i  -t = 1  so  x. X  T h i s c o n t r a d i c t s the m i n i m a l i t y o f  l  e Rx. 2  k  e Rx. i  + ... -i- Rx 2  + ... + Rx.  X  X  k  k .  S i n c e we have reached a c o n t r a d i c t i o n we may 2 conclude t h a t R/R Q.E.D. I f f o r a l l f i n i t e l y generated subrings R'  of  R,  cannot be homomorphically mapped onto a non-zero idem-  p o t e n t r i n g then c e r t a i n l y R e F F ' = Yi that R  R'  R e Yl  .  I t f o l l o w s then  i f and o n l y i f no f i n i t e l y g e n e r a t e d s u b r i n g o f  can be homomorphically mapped onto a non-zero  idempotent  ring.  R e c a l l that  j3  i s the upper r a d i c a l c l a s s  deter-  mined by the c l a s s o f a l l s u b d i r e c t l y i r r e d u c i b l e r i n g s w i t h idempotent h e a r t s .  The f o l l o w i n g lemma w i l l enable us t o  .0  - 131  jS *  prove t h a t  -  i s a radical  class.  5 . 3 . 2 LEMMA: If  S  i s a non-zero s i m p l e Idempotent r i n g then  there i s a f i n i t e l y generated subring  S'  of  S  which can  be homomorphically mapped onto a non-zero s u b d i r e c t l y i r r e d u c i b l e r i n g w i t h an idempotent h e a r t . Proof: (0) / S = S  Let  S ^ £  55 i n D i v i n s k y [7]  [7] t h e r e i s a • x e S (0) / S x S <f S  so  2  r  l  J  ' '"' k r  a n c i  be a s i m p l e r i n g .  x =  and so by Theorem 53 i n D i v i n s k y  such t h a t  S = Sx S . s  k  by the s e t in  S'  x e I  so - x  2  Then  S  such t h a t  2  (*)  1  T  \^' i> s  : x £ J} .  ducible with heart (*)  n  be the s u b r i n g of  (x,r^,  Z = ( J < s'  / 0 .  k S r . x s. . i=l 1  Let  ^  x  Thus t h e r e a r e elements  2  l > ''">  s  By Theorem .  S  • • • > \<Z ' S  Then  s'/I  Therefore  H  Choose  I  maximal  i s subdirectly Irre-  H = ((x) + I ) / I . £ I .  which i s g e n e r a t e d  If x 2  2  e I  / (0)  so  then by H  2  = H .  Q.E.D. An I n t e r e s t i n g c o n c l u s i o n t h a t f o l l o w s from t h i s lemma i s t h a t i f t h e r e i s a s i m p l e idempotent n i l r i n g t h e n t h e r e i s a s i m p l e idempotent n i l r i n g which i s the h e a r t o f  - 132 -  a f i n i t e l y generated n i l r i n g .  5.3.3  PROPOSITION: (1)  8 * < 0  and so  (2)  £ < 6^*  < Yl •  (3)  B^* 4 Yl  8 *  i s a radical  class.  i f and o n l y i f there i s a non-zero  . idempotent  simple  n i l ring.  Proof: (1)  Let  R efi.* .  If  R I |3  then  R  p h i c a l l y mapped onto a s u b d i r e c t l y a simple Idempotent h e a r t  is a contradiction is a local radical (2)  so  R e S  class.  r i n g with  R .  ^cp  i s not i n  B^ .  This  By Theorem 2 . 2 . 2  8 * ^cp  J  By Theorem 55 i n D i v i n s k y [ 7 ] no simple idempotent  ring  is i n £ .  so  Therefore  £ < 8 * . —  (3)  irreducible  But then by 5 - 3 . 2 some  S .  f i n i t e l y generated s u b r i n g of  can be homomor-  If  6 * £ 71  1  hear  a simple idempotent n i l r i n g .  8^*  < P  cp —  mapped onto a s u b d i r e c t l y  . r i n g with an idempotent  hence  S  then there i s a n i l r i n g which can be  homomorphically  idempotent  Clearly  cp  n i l ring  S / (0)  semi-simple s i n c e  H 4 (0) .  irreducible Clearly  H is  Conversely, any simple is  8^  semi-simple (and  8^* <_ B ) but i s i n Yl .  Q.E.D.  - 133 -  I n view o f 1.1.6 t h e f o l l o w i n g that  jS^* i s a s p e c i a l r a d i c a l  5.3.4  THEOREM: • A ring  R  theorem I m p l i e s  class.  i s /3^* . s e m i - s i m p l e i f and o n l y i f R  i s i s o m o r p h i c t o a s u b d i r e c t sum o f prime  0^*  semi-simple  rings.  Proof: S i n c e s u b d i r e c t sums o f s e m i - s i m p l e r i n g s a r e s e m i s i m p l e one d i r e c t i o n i s c l e a r . Conversely, l e t R  j3 * s e m i - s i m p l e r i n g .  be a  Cp  I s s u f f i c i e n t t o prove t h a t f o r a l l non-zero an i d e a l 0^*  l ( x ) such t h a t  semi-simple Let  x  l ( x ) and R / I ( x )  0 / x e R . Since  R ' such t h a t  R'  ( )R £ i ^*  of  X  J < R  I ' + (R' n J ) | S '  :a € A  J J  —  Then  U[J  X  R  and an I d e a l l '  R ' / l ' c o n t a i n s a non-zero s i m p l e idempo-  S ' / l ' . Notice that since  Let  i s a prime  there i s a  3  ( )  tent heart and  there i s  ring.  f i n i t e l y generated s u b r i n g of  x e R  It  then  S ' / l ' i s simple i f R' n J c i'  .  (*)  Z = [ J <? R : i ' + ( R ' n J ) ± S'} . L e t  be an a s c e n d i n g c h a i n o f i d e a l s : a e A} . By (*) , R' fl J  R ' n J c I ' hence  c I'  in Z  for a l l  I ' + R' P, J = I ' £ S'  T h e r e f o r e by Zorn's Lemma we may choose  and l e t  so  a e  A  .  J e Z .  I ( x ) ' maximal i n Z.  - 134 -  F i r s t we s h a l l prove t h a t  is 8 *  R/I(x)  semi-  cp  simple.  L/I(x)  Let  L ^ l(x) , i ' + R  Since 7  0 L + l(x)  n L o S  7  fl L „ — i ( x ) = R ' n i(x) p h i c a l l y mapped onto - — ^ j ^ " R  „  R  .  7  a  n  d  .  V  ' jr  c  is subdirectly irreducible Then  ( R fl L + 1 ) / 1 7  7  Now  L . R' n i(x) s i n c e by (*) R  7  H j-? L + 1  /  .  a  n  b e  i s n o t i n 3^* .  o  m  , _ and so R  I  ^  o  7  m  7  o  r  -  fl + L -f I '  S /I  so s i n c e  R /  Therefore  h  „  7  w i t h idempotent h e a r t  Thus  .  D  i s not i n 8  7  (R' n L + l ' ) / l ' L/I(x)  R  r-  S  Now  .  7  c  +  p' K n il Tl /( x\ ) r-c Ti'  R/I(x) .  be a non-zero i d e a l o f  7  .  8 * < 8  ,  and hence  X )  i s 8^* s e m i -  R/I(x)  simple. R/I(x)  Now we s h a l l prove t h a t Suppose t h a t R/I(x) I' s  and  L/l(x)  H/l(x)  a r e non-zero i d e a l s o f  LH c l ( x ) . By the maximal!ty o f  + ( R D L) 3 S 7  '  and  i s a prime r i n g .  s  /  and  7  + i  2  This i m p l i e s that d i c t i o n since  S  7  7  + ( R fl H) 3 S  _ (R' n c= I  7  7  L  + i UR  + (l(x) D R  7  7  7  .  Therefore  n H + i  7  ) _ i(x)n  R  7  R/l(x)  i s a prime  R  ) which i s a c o n t r a -  l(x)e Z .  Therefore r i n g and s i n c e  7  I  l(x) ,  8^*  semi-simple  £ l(x) , x { l(x) .  T h i s completes t h e p r o o f . Q.E.D.  7  - 135  -  5 . 4 LOCAL COMPLEMENTARY RADICAL CLASSES. d  Let class  ft  such  be a r a d i c a l c l a s s .  that:  (1) ft(A) n $ (A) = (0) (2)  If  f o r a l l rings  3" i s a r a d i c a l c l a s s such  3(A) n J (A) = (0)  .  We s h a l l denote  some r a d i c a l c l a s s e s  ft J  by  ft CRK(q£  A .  that  f o r a l l rings  then A n d r u n a k i e v i c [2] d e f i n e s J  I f there i s a r a d i c a l  A  then  t o be the complement of ) .  Notice that f o r  , C R H ( ^ ) may n o t e x i s t .  I n [ 2 ] A n d r u n a k i e v i c proves the f o l l o w i n g  5.4.1  3" <_ ft .  theorem.  THEOREM: If  ft  i s a hereditary  r a d i c a l c l a s s then  CRH(ft)'  e x i s t s and (i)  CRH(ft) = t h e upper r a d i c a l c l a s s determined by the class of a l l s u b d i r e c t l y i r r e d u c i b l e rings with in  (ii)  M .  R e CRH(ft) R  hearts  i f and o n l y i f e v e r y homomorphic image o f  i s ft s e m i - s i m p l e (such r i n g s a r e c a l l e d  strongly  ft s e m i - s i m p l e ) .  5.4.2  DEFINITION: Let  r a d i c a l class  be a r a d i c a l - c l a s s . ft  such  ( i ) ft(A) fl J (A)  I f there i s a l o c a l  that:  = (0)  f o r a l l rings  A .  - 136 -  (ii)  If 3  3(A) n jtf (A) = (0)  i s a l o c a l r a d i c a l c l a s s and  for a l l rings  A  then  3 <_ M .  t h e n tt i s t h e l o c a l complement o f We w i l l denote the l o c a l complement o f j$  5.4.3  by $ .  THEOREM: tt*  If  tt*  i s a l o c a l r a d i c a l c l a s s then  exists  and tt* = CRH(Ji*)* .  Proof: tt*  Notice that  i s h e r e d i t a r y so  CRH(W*)  We s h a l l prove t h a t  CRH(W*)*  s a t i s f i e s conditions  ( i ) and ( i i ) o f 5-4.2.  (1)  s a t i s f i e s c o n d i t i o n (A) so does  Since  CRH(tt*)  Suppose t h a t that both  A/B  and B  exists.  i s a r a d i c a l c l a s s which  B  CRH(W*)*,  i s an i d e a l o f a r i n g  a r e i n CRH(W*)*'.  a f i n i t e l y generated subring of s t r o n g l y tt* s e m i - s i m p l e t h e n  A . A  A and  L e t A'  I f A'  be  i s not  can be homomorphically  mapped onto a non-zero r i n g which i s n o t tt* s e m i - s i m p l e . Thus t h e r e i s a f i n i t e l y g e n e r a t e d s u b r i n g Such t h a t  L'  tt*  B  e CRH(tt*) . Thus  semi-simple.  (since  o f A'  can be homomorphically mapped onto  (0) / (L'/K') e tt* . S i n c e L  L'  L ' / L ' fl B  ^  A/B e CRH(M*)* , + B ~  I t follows that  ^  ± s  s  t  r  o  n  g  l  y  (L' ( I B ) + K' = L '  can be homomorphically mapped onto  - 137 -  L'/(L'  fl B + K )) .  p h i c a l l y mapped onto  Therefore  L' fl B  -—  = L'/K' e ft* .  n  Is a c o n t r a d i c t i o n since  K  ?  +  K  i / fl B'c B  can be homomor This  and hence i s i n  CRH(ft*) * (so no f i n i t e l y g e n e r a t e d s u b r i n g o f  L' n B  can be homomorphically mapped onto a non-zero r i n g i n ft*) . T h e r e f o r e e v e r y f i n i t e l y g e n e r a t e d s u b r i n g of A  i s strongly  A e CRH(ft*)*  .  a local radical (2)  Let  A  Let  R'  s e m i - s i m p l e so by 5-4.1 ( i i )  ft*  Then by P r o p o s i t i o n 2.2.1  CRH(ft*)*  is  class.  be a r i n g and l e t I = ft* ( A ) fl CRH(ft*)*(A) . be a f i n i t e l y g e n e r a t e d s u b r i n g o f  R' e ft* and Therefore of ft* .  R' e CRH(ft*)  R' = (0) Hence  since  I = (0)  so  I .  Then  R' = ft*(R') fl CRH(W*)(R ). /  CRH(1A*)  i s the compliment  so c o n d i t i o n ( i ) of 5.4.2 is^  satisfied. (3)  Suppose t h a t 3(A)  3" i s a l o c a l r a d i c a l c l a s s and  n ft*(A) = (0)  since  CRH(ft*)  f o r a l l rings  A .  Then  i s t h e compliment o f ft* .  3 = 3* <_ CRH(ft*)*  3 < CRH(ft*)  B u t then  so c o n d i t i o n ( i i ) o f 5.4.2 i s s a t i s f i e d .  T h e r e f o r e ft* = CRH(ft*)* . Q.E.D. I f f o l l o w s t h a t i f ft* i s a l o c a l r a d i c a l c l a s s then  Reft*  i f and o n l y i f e v e r y f i n i t e l y g e n e r a t e d  subring  - 138 -  of  R  i s strongly  tt*  semi-simple.  The f o l l o w i n g theorem shows t h a t t h e r e i s no need to d e f i n e elementary complements of l o c a l r a d i c a l  classes.  5 - 4 . 4 THEOREM: If  tt*  (tt*)  i s a l o c a l r a d i c a l c l a s s then  7  = tt*.  Proof: Let  tt*  Since Let  tt*  R .  and l e t J / I 7  7  <x> e tt* and s i n c e so so  i s a local class,  R e (W*)'  ted s u b r i n g o f R'  be a l o c a l r a d i c a l  J ' / l ' = (0) .  Let  and l e t R  7  class. (tt*) <_ ( t t * )  .  7  be a f i n i t e l y  genera-  R / l ' be a homomorphic image o f 7  =tt*(R /I ) . 7  7  R e (tt*)  If  x e J /l'  <x> € tt* .  7  Therefore, R  7  then  7  Thus  <x> = (0)  i s s t r o n g l y tt* s e m i - s i m p l e  Rett*. Hence, tt* = ( M * )  7  i s an e l e m e n t a r y  class. Q.E.D.  N o t i c e t h a t i f tt > ft then  CRH(tt) <_ CRH(ft) i f  t h e y b o t h e x i s t and tt <_ ft i f b o t h of these c l a s s e s e x i s t . Let  tt*  be a l o c a l r a d i c a l c l a s s and l e t R  r i n g which i s not i n tt* . ring  R  7  of  R  be a  Then some f i n i t e l y g e n e r a t e d sub-  i s not s t r o n g l y  tt*  semi-simple.  can be homomorphically mapped onto a r i n g  R"  Thus  such t h a t  R  7  - 139 -  tt*(R")  4  (0) .  C l e a r l y then  M*(R*) 4 R"  tt*(R ) fl tt*(R ) = (0), . T h e r e f o r e ,  since  R | tt* .  I t follows  t h a t tt* <_ f * . Suppose t h a t ft <_tt<_ 3" and a l l t h r e e c l a s s e s a r e radical classes. that rings  A l s o assume t h a t  I f JL  i s a l o c a l r a d i c a l c l a s s and  $ (R) n tt(R) = (0) f o r a l l r i n g s J (R) fl ft(R) c  and ft e x i s t and  1T(R) fltt(R)c T T ( R ) n 3T(R) = (0) f o r a l l  f = ft . Then R .  3  R  then  ^ ( R ) n tt(R) = (0) f o r a l l r i n g s  R  so  J <_ft= If . T h e r e f o r e tt e x i s t s and tt = IT = ft .  We s h a l l now i n v e s t i g a t e t h e l o c a l complements o f the r a d i c a l c l a s s e s we have been d i s c u s s i n g .  5.4.5 PROPOSITION: PI*  = {(0)}  Proof: We need o n l y show t h a t i f tt* i s a l o c a l c l a s s and tt* 4 ( ( 0 ) } ring  then  PI*(A)  n tt*(A) 4 (0)  radical f o r some  A . Suppose t h a t  0 4 R e tt* and t h a t tt* i s a l o c a l  r a d i c a l c l a s s . L e t 0 / x e R . Then <x> e tt* and so i s 2 2 ? <x>/<x> . I f <x> 4 <*> t h e n <x>/<x> can be homomorp  p h i c a l l y mapped onto a f i n i t e r i n g . non-zero  x e R , then  I f <x> = <x>  R e fi' so e v e r y f i n i t e l y  for a l l generated  - 140  subring i s f i n i t e .  In either  -  case we see t h a t t h e r e i s a  f i n i t e r i n g i n tt* . T h i s completes the p r o o f . Q.E.D.  Of  course the c l a s s .  ment o f the c l a s s  ((0)}  i s the l o c a l comple-  I t then f o l l o w s from 5 - 4 . 5  of a l l r i n g s .  t h a t i f tt i s a r a d i c a l c l a s s and tt >_ P I * and tt = {(0)}  .  In p a r t i c u l a r  then  tt  exists  then,  FT* = PC = W7&7* = ~& = T = ( £ ) =•"£ = ((0)} . R  5.4.6  R  G  G  PROPOSITION: A ,  P  and  Let  R  be a r i n g and l e t  0  FF  g  e x i s t and J = P = PP  g  =  .  Proof:  If  (0)  1/  then s i n c e  I e £'  I = e'(R) n FF (R) .  by 3 - 4 - 3  p h i c a l l y mapped onto an a l g e b r a i c f i e l d characteristic.  Since  p h i c a l l y mapped onto  Since  J  .  o f prime can be homomor-  K (see Theorem 4 6 i n D i v i n s k y  this i s a contradiction. = ((0)}  K  can be homomor-  I «3 FP (R) , FF (R) g g  Since a l l f i n i t e l y generated subrings of  £ ' fl PP  I  Therefore  K  I = (0)  S i m i l a r i l y one shows t h a t  [7]).  are f i n i t e f i e l d s so e' n P = {(0)}.  < PF_ , £' H J = ((0)} . Suppose t h a t tt i s a l o c a l r a d i c a l c l a s s and  - 141 -  34 4. £' . Then t h e r e I s a r i n g Thus  <x> 2  34 n PF  2 <x> ¥ <*> <  can be homomorphically mapped onto t h e t r i v i a l  <x>/<x> simple  <x> e 34 such t h a t  and so  <x>  zero r i n g .  ring  can be homomorphically mapped onto a 34 fl J  Thus  / {(0)} . I t follows  34 fl F 4 C ( 0 ) } £' = J = F =. F F  and Therefore,  -/ ( ( 0 ) }  that  f  Q.E.D.  Now i f $$ and & < F  i s any r a d i c a l c l a s s such t h a t  or. ^ <_ PF  then  = £' . T h i s i n c l u d e s a l l  of t h e r a d i c a l c l a s s e s l i s t e d i n Chapter I e x c e p t example : J = 71 = £ ' ) .  Of <_  FF ( f o r  I t a l s o includes the generalized  l o c a l and e l e m e n t a r y c l a s s e s d e t e r m i n e d by these r a d i c a l c l a s s e s ( f o r example :  J = J = £') and t h e l o c a l and g g-j_ . —  e l e m e n t a r y c l a s s e s d e t e r m i n e d by these r a d i c a l c l a s s e s ( f o r example : J * = J ' = £ ' ) . S i n c e any r i n g a ring  R,  R  can be embedded (as an i d e a l ) . i n  with i d e n t i t y i t i s clear that  1  £ s  of c o u r s e , i f <$  i s any r a d i c a l c l a s s and J  l  = { ( 0 ) } . So  >_ £  g  then  T = ((on . The r a d i c a l c l a s s plement; t h a t i s , PF the r a d i c a l  FP  does n o t have a l o c a l com-  does n o t e x i s t .  classes  ^n ® ^ p ^ n ^ =  S  :  P  i s  a  Prime]  To see t h i s  consider  - 142 -  where  i s the s e t of a l l p o s i t i v e i n t e g e r s Let  Since  R  be a r i n g and l e t I = ^ ( ) R  mapped onto a f i e l d  K  I e 3 , K e 3 , so n n ' 3  Theorem 46 i n D i v i n s k y  finite.  K .  I  can be homomorphically  i n £' (see 3«4.3)« K  i s a finite field. [7])  However, s i n c e B u t then (as i n v  F F ( R ) can be homomorphically  T h i s I s a c o n t r a d i c t i o n because  Therefore  I = (0)  so  a l l p o s i t i v e integers  e x i s t s then  Z c PP f o r n — U 3^ c FF so  n . But then  £ ' = (u 3 ) ' c (FF) ' = PP . n n  K is  3^ fl PP = ( ( 0 ) } .  I t f o l l o w s t h a t i f PP  £' fl PP  •  n  n  I e 3" <. £' , i f I 4 (0) ,  mapped onto  <_ n .  This i s impossible  since  contains a l l i n f i n i t e algebraic f i e l d s of f i n i t e  characteristic. Therefore,  PP  does n o t e x i s t .  We s h a l l now c o n s i d e r the l o c a l complements of e l e m e n t a r y r a d i c a l s which a r e  <_ PC .  5.4.7 PROPOSITION: (1)  R e PC a  l  x  +  a  2  i f and o n l y I f f o r a l l x e R , x  2 k + ••' + a^x = 0  such t h a t not d i v i d e (2)  p  f o r some i n t e g e r s  divides a l l  for i > 1  a^ .  I f S 4 gf then  3 71(S) = F C p  p  .  but  a^,...,aj p  does  - 143 -  (3)  Re  3: Yl (0)  i f and o n l y i f f o r a l l x e R ,  p  a-^x + . .. + a^x^ = 0  (4)  such t h a t  p  If S 4 0  then  f o r some i n t e g e r s  does n o t d i v i d e  l'  j (5) X J !  but  s  p  u  c  that  h  p  f o r some i n t e g e r s  does n o t d i v i d e  divides a l l a  i  f o r which  a . • f o r some  i 4 j •  FC = PC . P P  ^n(^) •  (6)  (7)  k  a  .  R e 3 (S) i f and o n l y i f f o r a l l  x e R , a^x +• ... + a^x^" = 0 a  a^, . .., a ^  T^sJ = ff (z ) i f +  p  Proof: (l)  R- i s a r i n g such t h a t f o r a l l x e R ,  Suppose  a^x + .. . + k a  such t h a t  p  x k  0  =  f o r some i n t e g e r s  does n o t d i v i d e  a^  a^, . .  but p  a^  divides  a^  for a l l i > 1 . If such t h a t a  l'  p  a  FC (R) 4 (0) then t h e r e i s a p ' '  x / 0 ic  s  u  c  n  b u t px = 0 . B u t t h e r e a r e i n t e g e r s that  does not d i v i d e  Then  a^x = 0  x e PC (R) P  p  a^  and s i n c e  divides and p  T h i s i s a c o n t r a d i c t i o n so  a^  f o r i > 1 but  a-^x + .. . + a ^ x = 0 . k  does n o t d i v i d e P C ( R ) = (0) . p  t h a t every f i n i t e l y g e n e r a t e d s u b r i n g o f  R  a^ , x = 0. I t follows i s strongly  - 144 -  PC  P  s e m i - s i m p l e so Let  R e PC  R e FC  P  and l e t x e R .  Then  <x>  is  P strongly are l' a  FC  integers '''' k a  s e m i - s i m p l e so  p  a.^ . a  n  d  x  2  ( a ^ - l ) x -8- a x divide  a  l  X  +  Thus t h e r e  a ^ such t h a t p d i v i d e s '"' k ' Therefore +  + ... + a^x  2  not  =  <x> = <px> . a  k  x l C  = 0  and c l e a r l y  p  does  a^ - 1 . T h i s completes the p r o o f o f ( l ) . (2),  ( 3 ) , (4) The p r o o f s f o r (2), (3), (4) a r e  i n a l l r e s p e c t s s i m i l a r t o the p r o o f o f ( l ) . We have a l r e a d y seen t h a t Suppose 0 4 x e R' .  Let  I f FC(<x>) 4 ( 0 )  FC(<x>) = ( 0 ) there i s a  <_ F C  p  R £ FCp .  qy = 0  such t h a t  FC  y e <x>  be h o m o m o r p h i c a l l y mapped onto q 4 p . <x>  Vl(<x>) = 0  If  R' = R/FC (R) . L e t p  subring of  R  <x>  7£(<x>) 4 ( 0 )  If  <y> = 0°° C^  so  <y>  can  f o r some prime  t h e n as i n .the p r o o f o f 3 - 3 . 2  can be homomorphically mapped onto a f i n i t e  of c h a r a c t e r i s t i c  ye  q / p . If  7L(<x>) .  such t h a t  .  then t h e r e i s a  f o r some prime  consiser  p  q 4 P •  field  Thus I n any case t h e r e i s a  w h i c h i s g e n e r a t e d by one element and  which can be homomorphically mapped onto a r i n g o f prime characteristic If  q 4 P • q  I s a prime and  q 4 P  then a r i n g o f  - 145  characteristic  q  -  is in  PC  (take  a l  p for  1 > 1).  element  Therefore, i f  x e R  such t h a t  R £ FC  <x>  n  = q  a. i  t h e r e i s an  p  i s not s t r o n g l y  s e m i - s i m p l e . Thus R £ F C . I t f o l l o w s t h a t PC  =0  FC p  p  = FC  3  P (6),(7) The p r o o f s f o r (6) and of  P •  (7) a r e s i m i l a r t o the p r o o f  (5). Q.E.D.  I n view o f the f o l l o w i n g theorem P r o p o s i t i o n c o m p l e t e l y determines  the l o c a l complements of e l e m e n t a r y  r a d i c a l c l a s s e s which a r e  5.4.8  <_ PC .  THEOREM: If  34 = ©  5.4.7  [tt  34 <_ FC  : p e S)  , 34  i s an e l e m e n t a r y r a d i c a l c l a s s  i s the r e p r e s e n t a t i o n of  3-4.14 then tt = n(¥  tt  and  given i n  : p e S} .  Proof: Let  34 = ^  (34  p  : p e S}  be-an'elementary  Since  p e S .  34 > 34 f o r a l l p e S , tt < — p ^ T h e r e f o r e tt <_ n(¥ : p e S] . Suppose  is strongly is strongly  R e n(34 : p e S} p  and  • 34 s e m i - s i m p l e f o r a l l p e S p  34 s e m i - s i m p l e .  Thus  R e 34 .  tt P  x e R .  radical. for a l l  Then  so c l e a r l y  <x> <x>  - 146 -  T h e r e f o r e ft = n{)(  : p e S} Q.E.D.  By v i r t u e o f 5.4.7 and 5.4.8 Since  &' = e'  (Z ) : p +  : p  o f ' = + f.3"  Since  i s a prime} = Jl  : P  i s a prime}  = n[3~^TpT : p  J'  i s a prime} = e' .  p  &  Since  i s a prime}  +  = fl(3P (z )  n PC = © ( 3 l t ( Z ) +  R  <&  p  : p  n PC = n(3: ^(Z ) : p +  R  we can c o n c l u d e :  p  i s a prime} i s a prime} = {(0)} .  We have seen t h a t ft' = ft  and  = e' .  In fact,  this implies that:  5.4.9  PROPOSITION: F o r any l o c a l r a d i c a l c l a s s  31*  (1)  34* fl Yl = {(0)}  i f and o n l y i f  (2)  34* n £' = {(0)}  I f and o n l y i f  34* <_ t' . 34* < %  .  Proof: Let and  &'  .34* be a l o c a l r a d i c a l c l a s s .  34* , YL  a r e h e r e d i t a r y i t f o l l o w s t h a t i f 34* n Jl = ((0)}  then ft* 1 TJT = ft* < T  Since  =  •  and i f ft* fl £' = ((0)}  then  - 147 -  34* n |  Clearly i f 34* n &' 4 { ( 0 ) }  then  4 ( ( 0 ) ) then  34* £  34* £ &'  and i f  . Q.E.D.  5 - 5 A REPRESENTATION OF  &'  AS THE INTERSECTION OF RADICAL CLASSES,  Many o f t h e r a d i c a l c l a s s e s  which we have d i s c u s s e d  can be r e p r e s e n t e d as the i n t e r s e c t i o n of two o t h e r r a d i c a l classes.  The f o l l o w i n g diagram i l l u s t r a t e s the s i t u a t i o n .  ILLUSTRATION 5 R e c a l l that  £ = Yi fl 3 . 8 . *  , FI*=  RFC  and  J** = £ fl F I * .  I t seems n a t u r a l 34 £ e'  such t h a t  to ask i f there i s a r a d i c a l  F I * fl 34 = e'  .  class  We s h a l l see t h a t no such  c l a s s e x i s t s i f we demand t h a t i t be a l o c a l c l a s s .  However,  - 148 -  we s h a l l prove t h a t  CRH( Jl ) fl F I * = &' . g Me s h a l l b e g i n w i t h t h e f o l l o w i n g easy lemmas  5 . 5 . I LEMMA: If  i s Y | s e m i - s i m p l e then g  R e F I * and R  R e e' .  Proof: Let  R  be a  Let  x e R . Then  <x>  i s a commutative Wedderburn r i n g . Thus  <x> = <x>  2  <x>  s e m i - s i m p l e r i n g which i s i n  <x>  i s f i n i t e and  i s a f i n i t e direct  . Therefore  /£(<x>) = (0)  FI*  so  sum o f f i e l d s so  R e &' . Q.E.D.  5.5.2 LEMMA:  (CRH(7l ))* = C' = (CRH( 7?^)) ' . g  Proof: If Yl  R e S'  semi-simple.  then e v e r y s u b r i n g o f R  Therefore  g <x>/<x>  £' <_ (CRH( Yl ))* g  i s strongly .  Suppose  R e (CRH( 7? ) ) ' . L e t 0 4 x e R .  is  s e m i - s i m p l e so  VI g  o  <x> = <x>  .  Then  Therefore  R e &' . Therefore  t' <  (CRH(Yl  ) ) * <_ ( C R H ( * y ^ ) ) ' < &' . Q.E.D.  - 149 -  N o t i c e t h a t i n t h e p r o o f of 5 . 5 . 2 we a c t u a l l y t h a t i f <x> e CRH(  (CRH(71 )) g  <e  gi  )  then  <x> e £ .  show  Thus  •  5 . 5 . 3 THEOREM: CRH( 77 ) n P I * = e' c l a s s ' such t h a t . ft n F l * = £'  and i f ft i s a l o c a l r a d i c a l then ft <_ CRH( $  ) so  ft = e' . Proof: As we saw i n 5 - 5 . 2 , &' <. CRH(  fl  )  If  R e CRH( $  CRH(V|  Suppose t h a t ft n P I * = e' .  a l o c a l class nilpotent.  ) n FI*  ft  )  i s Yl  n PI* =  I f ft i CRH(Y| )  <x>  semi-simple  &' .  i s a l o c a l r a d i c a l c l a s s such t h a t  there i s a r i n g  finite n i lring  R  s  i s n o t s t r o n g l y Y|  Then  then  g R e £' .  Therefore  R  Thus  nF I * .  so by Lemma 5-5-1*  such t h a t  ) •  £' < CRH(W  then t h e r e i s a r i n g semi-simple.  <x> e ft such t h a t  can be homomorphically  <x'> . The r i n g  Reft  S i n c e ft i s <x> i s  mapped onto a  <x'> e P I * D ft = e' .  T h i s i s a c o n t r a d i c t i o n so ft <_ CRH(y| ) N o w ft = ft* <_ (CRH(Y| ) ) * = e '  by-Lemma 5 - 5 - 2 .  T h e r e f o r e ft = &' . Q.E.D.  - 1 5 0 -  Of course numbers  CRH(Yjg) 4  since the r i n g of r a t i o n a l  Q e CRH( Y| ) . T h i s example a l s o show t h a t  CRH( V / ) i PC . On the o t h e r hand, 7J  fl FC  { ( 0 ) } so  PC 4 CRH(Y| ) • Notice that  CRH(Y?  6  )  0 FC ^ S' .  F o r an example  c o n s i d e r any f i e l d o f f i n i t e c h a r a c t e r i s t i c which i s n o t algebraic. We may sum up the r e l a t i o n s h i p s between these r a d i c a l c l a s s e s i n t h e f o l l o w i n g diagram.  ILLUSTRATION 6  - 151  -  . 6 SEMI-SIMPLE RADICAL CLASSES. I n t h i s s e c t i o n we s h a l l c h a r a c t e r i z e those classes  ft*  classes.  local  which a r e b o t h r a d i c a l c l a s s e s and s e m i - s i m p l e  We s h a l l see t h a t a l l r a d i c a l c l a s s e s which a r e  a l s o semi-simple classes are i n f a c t l o c a l r a d i c a l c l a s s e s ( i n d e e d , e l e m e n t a r y ' c l a s s e s ) so we s h a l l b e g i n w i t h t h e more general  problem.  5 . 6 . 1 LEMMA:If  ft  i s a c l a s s of r i n g s such t h a t s u b d i r e c t sums  of r i n g s i n ft a r e i n ft and such t h a t t i o n (A) then  ft  satisfies  ft  i s strongly hereditary.  ft  be a c l a s s of r i n g s such t h a t  condi-  Proof: Let  sums of r i n g s i n ft a r e i n ft and such t h a t  subdirect  ft  satisfies  condition (A). Let  integers.  Reft  and  S  be a s u b r i n g o f  R .  Now the ( d i s c r e t e ) d i r e c t ..sum S(R^ : i e Z } i s  an i d e a l of the d i r e c t p r o d u c t (complete d i r e c t sum) +  {R  ±  : i e Z } .  of c o p i e s  of  R  A ( S ) + E(R  : i e Z} +  ±  and hence i s i n ft , so  i s a subdirect  sum  - 152 -  + S[R. : 1 e Z } e ft . SCR : i e Z }  A(S)  S ~ A(S) ~  +  +  ±  Q.E.D. U s i n g a theorem o f .Amitsur [ l ] which s t a t e s e v e r y r i n g i s a homomorphic image o f a s u b d i r e c t  that  sum o f t o t a l  m a t r i x r i n g s o f f i n i t e o r d e r over the r i n g o f a l l i n t e g e r s , Armendariz i n [ 5 ] proves t h a t i f a h y p e r n i l p o t e n t class rings.  ft  5.6.2  i s a s e m i - s i m p l e c l a s s , then ft c o n t a i n s a l l  R e c a l l that a hypernilpotent  hereditary  radical  r a d i c a l class i s a  r a d i c a l c l a s s which c o n t a i n s a l l n i l p o t e n t  rings.  THEOREM: I f ft i s a s e m i - s i m p l e r a d i c a l c l a s s and ft £ &'  then ft i s the. c l a s s o f a l l r i n g s .  Proof: Let  ft  be a s e m i - s i m p l e r a d i c a l c l a s s .  then t h e r e i s an R e f t  and an  x e R  such t h a t  S i n c e ft i s a s e m i - s i m p l e c l a s s s u b d i r e c t ft are i n ft so by 5 . 6 . 1  (P),  C  w  2  <x>/<x>  e ft . T h e r e f o r e  lower r a d i c a l c l a s s determined by a hypernilpotent  <x> 4 <x> •  sums o f r i n g s i n 2  <x> e ft . Now  z e r o r i n g on a c y c l i c group and satisfies  I f ft ^ £'  <x>/<x>  is a  e ft . S i n c e ft  6 <_ ft s i n c e  6 = the  [C ] . T h e r e f o r e ft i s a  r a d i c a l c l a s s so by t h e p r e c e d i n g remarks  ft i s t h e c l a s s o f a l l r i n g s .  Q.E.D.  -  153  -  We can now prove t h a t a l l s e m i - s i m p l e r a d i c a l c l a s s e s must be e l e m e n t a r y  classes.  5 . 6 . 3 LEMMA: If  ft  i s a s e m i - s i m p l e r a d i c a l c l a s s then ft = ft'.  Proof: L e t ft be a s e m i - s i m p l e r a d i c a l c l a s s . 5-6.1,  ft  Then by  <_ ft' .  If  ft  i s the c l a s s o f a l l r i n g s then c l e a r l y  If  ft  i s n o t the c l a s s of a l l r i n g s then by 5 . 6 . 2 ,  ft = ft .  ft < _ e' 3.4.3 F  so R  &'<&'.  where each  characteristic. Reft'  d i r e c t sum) that  F„  i s an a l g e b r a i c f i e l d o f prime  L e t j3 e A . Then f o r a l l x e F^ , <x> e ft  . T h e r e f o r e the d i r e c t p r o d u c t  (complete  A =~j]"(<x> : x e P^} e ft <_ £' . L e t y e A  y(x)= x  Therefore  (0) 4 R e ft' . Then by  i s i s o m o r p h i c t o a s u b d i r e c t sum o f f i e l d s  : a e A  since  Suppose  F^  . By 3 . 4 . 6 <y>  for a l l x e must be f i n i t e .  g e n e r a t e d by one element each  such  i s finite.  Since f i n i t e f i e l d s are F  i s i n ft and s i n c e s u b -  a d i r e c t sums o f r i n g s i n ft a r e i n ft , R e ft .  Hence  ft' <_ ft . T h e r e f o r e ft = ft' so ft i s an e l e m e n t a r y c l a s s . Q.E.D.  - 154  -  5 - 6 . 4 LEMMA: If  3  Is a s t r o n g l y h e r e d i t a r y f i n i t e s e t of f i n i t e  f i e l d s then a r i n g fields i n of  R  3  R  i s i s o m o r p h i c t o a s u b d i r e c t sum o f  i f and o n l y i f e v e r y f i n i t e l y g e n e r a t e d  subring  i s i s o m o r p h i c t o a f i n i t e d i r e c t sum of f i e l d s i n  3 .  Proof: Let  3  finite fields.  be a s t r o n g l y h e r e d i t a r y f i n i t e s e t o f Then i f F e 3  x ^^ = 1  such t h a t  Assume t h a t Q  S P  e 3  a  R  Then I f  a  f i n i t e l y g e n e r a t e d s u b r i n g of by 3 - 4 . 4 Choose  R  = A  7  x  © ... © A  k  A}  R.  =  x  I  : a e A  (0) .  Let  By 3 - 4 - 5  where t h e  A  ±  N  = x .  such t h a t  R'  be a  R' e £' . are f i n i t e  a. e R, such t h a t I  a. £ I _ i p\  f o r some  1  <a.> i  x e F e 3 ,  has i d e a l s  n(I : a e  and  n(P)  f o r a l l x e F - Let  n  N ="["[{n(F) : F e 3} + 1 .  R/I  t h e r e i s an i n t e g e r  n I  p. i  = (0)  .  <a. > = A. . Then a. / 0 i i I j3. e A . Now <a.> fl I <1 <a.> * i I I  i s Isomorphic t o a s u b r i n g of  fields,  so so  a  Therefore  Then  A. ~ < , > = (<a."> + I l I i p )/I . p . * i 'I x  a  F  s  .  Since  Q  3  a  i s strongly  'I  hereditary in  R'  i s i s o m o r p h i c t o a f i n i t e d i r e c t sum o f f i e l d s  3 . C o n v e r s e l y , assume t h a t e v e r y f i n i t e l y g e n e r a t e d  subring of in  3 .  R  Then  i s i s o m o r p h i c t o a f i n i t e d i r e c t sum of f i e l d s x  = x  for a l l x e R  so  R e Z  .  Thus, by  -  3-4.3 nfl  -  155  there are i d e a l s  I : a e A o f R such a and R/I i s a f i e l d of f i n i t e  : a e A] = (0)  c h a r a c t e r i s t i c f o r a l l a e A . B u t then f i n i t e f i e l d since f o r e , f o r each  x  x  s  o  a  R  /l  . B u t then i s a  l  s  o  m  o  r  must be a  a  for a l l x e R .  a e A , t h e r e i s an  (<x > + I ) / I = R/I image o f < >  - x = 0 e I  N  R/*  x  e R  Q  R/I„  P i n  c  t  o  There-  such t h a t  i s a homomorphic a  field i n 3 . Q.E.D.  5 . 6 . 5 THEOREM: If  ft  i s a c l a s s o f r i n g s which i s n o t t h e c l a s s  of a l l r i n g s then t h e f o l l o w i n g a r e e q u i v a l e n t : (1) (2)  ft  i s a semi-simple r a d i c a l c l a s s .  There i s a f i n i t e s e t o f primes  T  and f o r each  a f i n i t e C.U.D. s e t o f p o s i t i v e i n t e g e r s R e ft • i f and o n l y i f R  S  p e T  such t h a t  i s isomorphic t o a subdirect  sum o f f i e l d s i n (P ^ : p e T and a e S •) . P There i s a f i n i t e s e t o f primes T and f o r each y  (3)  a f i n i t e C.U.D. s e t o f p o s i t i v e i n t e g e r s  S P  p e T  such t h a t  R = © C 3 p ( S ) : p e T} . p  Proof: L e t ft be a c l a s s o f r i n g s which i s n o t t h e c l a s s of a l l r i n g s , Lemma 5 - 6 . 4 i m p l i e s t h a t ( 2 ) and ( 3 ) a r e e q u i v a l e n t .  - 156 -  Assume t h a t ft i s a s e m i - s i m p l e r a d i c a l Then by 5 . 6 . 2 and 5 - 6 . 3 & and ft <_ e ' 3  p e T , l-e S  integers . Let R =  p  semi-simple c l a s s so c e r t a i n l y S P  T  3-4.14,  f o r some s e t o f primes  S  sets of p o s i t i v e  t h a t each  i s an e l e m e n t a r y r a d i c a l c l a s s  . T h e r e f o r e , by Theorem  ft = © ( p ( p ) : P e T]  S  f o r each  p  TR  F  p  R €•£' , R e FC  J u s t as i n 5 - 6 . 3 we see TTfF  must be f i n i t e by c o n s i d e r i n g  :a e S } . P  p 3  T  and C.U.D.  : P e T] . S i n c e ft i s a  R e ft <_ £' . Because must be f i n i t e .  T  p e T . F o r each  C o n v e r s e l y , assume t h a t ft = © ( p ( S where  class.  i s f i n i t e and f o r each  C.U.D. s e t o f p o s i t i v e  integers.  ) : p e T]  p e T ,  S Is a finite p Then ft i s a r a d i c a l c l a s s .  S i n c e ft i s an e l e m e n t a r y c l a s s ft s a t i s f i e s condition (E). Suppose t h a t e v e r y non-zero i d e a l o f a r i n g  R can  be homomorphically mapped onto a non-zero r i n g i n ft . by 5 . 6 . 4 e v e r y i d e a l o f R { F : p e T , a e S } P  Then  can be mapped onto a f i e l d i n  . From t h e p r o o f o f Theorem 4 6 i n  p  D i v i n s k y [7] we see t h a t t h i s i m p l i e s t h a t to a s u b d i r e c t sum o f f i e l d s i n (F P  R  i s isomorphic  : p e T , a e S] .  Then, by 5 . 6 . 4 a g a i n , R e ft* = ft . Thus ft s a t i s f i e s c o n d i t i o n (F) . T h e r e f o r e ft i s a s e m i - s i m p l e r a d i c a l  class. Q.E.D.  - 157  -  The r e l a t i o n s h i p s between the l o c a l r a d i c a l c l a s s e s which we have d i s c u s s e d can be i l l u s t r a t e d i n the f o l l o w i n g diagram.  ILLUSTRATION 7  - 158  -  BIBLIOGRAPHY [1]  S. A. A m i t s u r , "The i d e n t i t i e s o f P. I . - r i n g s " , P r o c . Amer. Math. Soc. 4 , 2 7 - 3 4  [2]  V. A n d r u n a k i e v i c , " R a d i c a l s i n a s s o c i a t i v e r i n g s I " , Mat. Sb., 4 4 , 1 7 9 - 2 1 2  [3]  (1958).  , " R a d i c a l s i n a s s o c i a t i v e r i n g s I I " , Mat. Sb.,  [4]  (1953).  5 5 , No. 3 ( 9 7 ) ,  329-46  (1961).  V. A n d r u n a k i e v i c and J u . M. R j a b u h i n , "Rings n i l p o t e n t e l e m e n t s , and c o m p l e t e l y s i m p l e  without  ideals",  S o v i e t Math. D o k l . , V o l . 9 , No. 3 , 5 6 5 - 6 8 ( 1 9 6 8 ) . [5]  E. D. A r m e n d a r i z ,  "Closure p r o p e r t i e s i n r a d i c a l theory",  Pac. J . Math. 2 6 , 1-8 [6]  (1968).  R. B a e r , " R a d i c a l I d e a l s " , Am. J . Math., 6 5 ,  537-68  (1943). [7]  N. J . D i v i n s k y , R i n g s and R a d i c a l s , (Toronto: of Toronto  [8]  1965).  L. P u c h s , A b e l i a n Groups, (Pergamon P r e s s , I n c . , New York,  [9]  Press,  i960).  D. Goldman, " H i l b e r t Rings and the H i l b e r t s a t z " , Math. Z e i t . 54, 136-140,  [10]  Nullstellen-  (1951).  I . N. H e r s t e i n , "Theory of R i n g s " , U n i v e r s i t y o f Chicago Math. L e c t u r e Notes,  [11]  University  (1961).  N. Jacobson, S t r u c t u r e o f R i n g s , (Am. Math. Soc. C o l l .  - 159  Publ. 37, [12]  -  1964).  I . N i v e n and H. S. Zuckerman, An I n t r o d u c t i o n t o the Theory of Numbers, (New York: Inc.,  [13]  John W i l e y and Sons,  I960).  J u . M. R j a b u h i n , " S e m i s t r i c t l y h e r e d i t a r y r a d i c a l s i n p r i m i t i v e c l a s s e s o f r i n g s " , Akad. Nauk. Moldav. SSR, Kishinev,  [14]  1965-  G. T h i e r r i n , "Sur l e s i d e a u x complement p r e m i e r s  d'un  anneaux guelconque", B u l l . Acad. Roy. B e l g . , 4 3 ,  124-  32 (1957). [15]  0. Z u r i s k i and P. Samuel, Commutative A l g e b r a , V o l . I ( P r i n c e t o n , N. J . : Van N o s t r a n d , 1 9 5 8 ) .  

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