UBC Theses and Dissertations

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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 C L A S S E S OP R I N G S b y P A T R I C K NOBLE STEWART B . A . , U n i v e r s i t y o f B r i t i s h C o l u m b i a , I965 M . A . , U n i v e r s i t y o f C a l i f o r n i a a t B e r k e l e y , 1966 A T H E S I S SUBMITTED I N P A R T I A L F U L F I L M E N T OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OP PHILOSOPHY i n t h e D e p a r t m e n t o f MATHEMATICS We a c c e p t t h i s t h e s i s a s c o n f o r m i n g t o t h e r e q u i r e d , s t a n d a r d ' T f f i T t JN I V E R S I T Y O F ' B K l T l S H COLUMBIA J u l y , 1969 1 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced d e g r e e a t the U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and S t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by the Head o f my Department or by h i s r e p r e s e n t a t i v e s . It i s u n d e r s t o o d t h a t c o p y i n g o r ' . p u b l i c a t i o n o f t h i s thes, is f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f The U n i v e r s i t y o f B r i t i s h Co lumbia V a n c o u v e r 8, Canada Supervisor: N. J. Divinsky i i ABSTRACT For any cardinal number K >_ 2 and any non-empty class of rings ft we make the following d e f i n i t i o n s . The • class ft(K) i s the class of a l l rings R such that every subring of R which i s generated by a set of c a r d i n a l i t y s t r i c t l y less than K i s i n ft . The class Rg(K) x S ^ e class of a l l rings R such that every non-zero homomorphic image of R contains a non-zero subring i n ft which i s generated by a set of c a r d i n a l i t y s t r i c t l y less than K . Several properties of the classes R g ( K ) a n d are determined. In p a r t i c u l a r , conditions are s p e c i f i e d which imply that ft(K) i s a r a d i c a l class or a semi-simple cla s s . Necessary and s u f f i c i e n t conditions that the class IT of a l l Rg(K) semi-simple rings be equal to 3"(K) are given. The classes ft(K) and ft / l M when K = 2 or v 1 g(K) K = £| are considered i n d e t a i l f o r various classes ft (including the cases when ft i s one of the well-known r a d i -c a l c l a s s e s ) . In a l l cases when ft i s one of the well-known r a d i c a l classes i t Is shown that ft(2) and ft(}»^o) are r a d i c a l classes and whenever they contain a l l nilpotent rings they are shown to be s p e c i a l r a d i c a l classes. Those r a d i c a l classes ft(2) which are contained i n PC (R € FC i f and only i f f o r a l l x e R , x i s torsion) are characterized. i i i Let ft be any r a d i c a l c l a s s . The largest r a d i c a l class M(H0) ( i f one exists) such that W(H0)(R) n a(R) = ( 0 ) f o r a l l rings R i s defined to be the l o c a l complement of ft and i s denoted by ft . I f ft = R(K0) then the l o c a l complement ft exists and ft = ft(2) . The l o c a l complements of a l l rad i c a l s discussed are determined. We are able to apply some of these re s u l t s i n order to c l a s s i f y those classes of rings which are both semi-simple and r a d i c a l classes. i v TABLE OP CONTENTS Page INTRODUCTION 1 CHAPTER I PRELIMINARIES 1.1 Radical Theory. • • - 4 1.2 Rings Without Nilpotent Elements 9 CHAPTER I I K-CLASSES AND GENERALIZED K-CLASSES 2.1 K-Classes 12 2.2 Local Classes 16 2- 3 Elementary Classes 32 2.4 Generalized K-Classes 38 CHAPTER I I I ELEMENTARY RADICAL CLASSES 3.1 The Elementary Radical Classes tr , trR and PC. 57 3.2 The Elementary Radical Classes $' and £>' . 62 3- 3 Classes f o r which H' = "fl 65 3.4 Elementary Radical Classes which are <_ PC . . 67 CHAPTER IV GENERALIZED ELEMENTARY AND LOCAL RADICAL CLASSES 4.1 Absorbent Cardinal Numbers 85 4.2 Generalized Radical Classes which are >_ y\ 92 4.3 Generalized Radical Classes which are <_ PC . 98 4.4 The Generalized Radical Class e 107 s l CHAPTER V LOCAL RADICAL CLASSES 5.1 The Local Radical Classes and £ . 114 5.2 The Local Radical Classes J*, and PI* . 118 5 . 3 Local Radical Classes M for which £ <_ 3* <_7| 129 5.4 Local Complementary Radical Classes . 135 5 . 5 A Representation of &' as the Intersection of Radical Classes . 147 5 . 6 Semi-Simple Radical Classes 151 BIBLIOGRAPHY I58 V o ILLUSTRATIONS Page 1. Elementary Radical Classes 84 2. Generalized Radical Classes . . . . . . . 113 3. Local Radical Classes . . . . . 125 4. The Classes J , $* , j' and J . 128 5. Intersections of Local Radical Classes 14-7 6. Intersections of Local Radical Classes and CRH(y)e.) ' 150 7. Summary of Local Radical Classes . 157 v i A CKNOWLEDGEMENTS The author wishes to thank h i s r e s e a r c h s u p e r v i s o r , Dr. N. J . D i v i n s k y , f o r advice and encouragement d u r i n g the p r e p a r a t i o n of t h i s t h e s i s . The author a l s o wishes to thank h i s wife f o r t y p i n g the f i r s t d r a f t of the t h e s i s , and Mrs. Monlisa Wang f o r t y p i n g the f i n a l d r a f t . The f i n a n c i a l support of the N a t i o n a l Research C o u n c i l of Canada i s g r a t e f u l l y acknowledged. - 1 -INTRODUCTION The purpose of t h i s t h e s i s i s to I n v e s t i g a t e r a d i -c a l c l a s s e s and semi-simple c l a s s e s which are determined by c o n d i t i o n s on f i n i t e l y generated s u b r i n g s . A c l a s s of r i n g s ft w i l l be c a l l e d a l o c a l c l a s s i f a r i n g R e f t i f and onl y i f every f i n i t e l y generated s u b r i n g of R s a t i s f i e s some s p e c i a l c o n d i t i o n . S i m i l a r i l y , a c l a s s of r i n g s ft i s an elementary c l a s s i f a r i n g R e f t i f and onl y i f every sub-r i n g of R generated by one element s a t i s f i e s some s p e c i a l c o n d i t i o n . L e t us c o n s i d e r some examples. C l e a r l y the c l a s s of a l l commutative r i n g s i s a l o c a l c l a s s s i n c e a r i n g R i s commutative i f and o n l y i f every f i n i t e l y generated sub-r i n g of R i s commutative. T h i s c l a s s of r i n g s Is not an elementary c l a s s . The c l a s s of a l l n i l r i n g s i s an elementary c l a s s s i n c e a r i n g R i s n i l i f and onl y i f f o r a l l x e R the s u b r i n g generated by x i s n i l . We s h a l l prove t h a t the c l a s s - o f a l l Jacobson r a -d i c a l r i n g s i s not a l o c a l c l a s s . N o t i c e t h a t the d e f i n i -t i o n of a Jacobson r a d i c a l r i n g ( f o r a l l x e R there i s a y e R such t h a t x + y + xy = 0 ) i n v o l v e s the e x i s t e n t i a l q u a n t i f i c a t i o n of a r i n g element. On the other hand the de-f i n i t i o n of a n i l r i n g i n v o l v e s o n l y the 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. T h i s i l l u s t r a t e s what we would - 2 -n a t u r a l l y expect : that classes of rings which are defined by conditions involving only universal 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 classes whereas classes of rings which are defined by conditions involving e x i s t e n t i a l quanti-f i c a t i o n of r i n g elements would not be l o c a l classes. For example, we would expect that the class of a l l rings s a t i s -f y i n g a given set of polynomial i d e n t i t i e s would be a l o c a l c l a s s . In Chapter I I we consider some general r e s u l t s about l o c a l classes. In p a r t i c u l a r , we specify several con-diti o n s under which l o c a l classes are r a d i c a l classes or semi-simple classes. The remainder of the thesis i s devoted to a consi-deration of s p e c i f i c l o c a l r a d i c a l classes and s p e c i f i c l o c a l semi-simple classes. Any class of rings ft determines a l o c a l and an elementary class (the class of a l l rings such that every f i n i t e l y generated subring (subring generated by one element) i s i n ft). We consider the l o c a l and elementary classes de-termined by the well-known r a d i c a l classes. A l l of these classes are r a d i c a l classes and those which contain a l l n i l -potent rings are shown to be sp e c i a l r a d i c a l classes. In this and i n other ways we arr i v e at several new r a d i c a l classes. A l l those which are elementary classes and which are contained i n FC (FC i s the class of a l l rings R such that f o r a l l x e R , x i s torsion) can be characterized as - 3 -"sums" of c e r t a i n simple elementary r a d i c a l c l a s s e s . I n some cases we are able to 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 c o n d i t i o n s . Any c l a s s of r i n g s which i s closed under homomor-phic 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 elementary and a l o c a l semi-simple c l a s s . These classes are i n v e s t i g a t e d i n Chapter IV. Fo l l o w i n g Andrunakievic [2] we define l o c a l com-plementary r a d i c a l s and determine the l o c a l complements of the r a d i c a l s which are discussed. F i n a l l y we are able to apply some of our r e s u l t s i n - o r d e r to c l a s s i f y a l l c lasses which are both semi-simple and r a d i c a l c l a s s e s . - 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 N c R , <x1} xN> = < C x2' ' • • •» ^ * (iii) If S c R , (S)R = ideal of R generated by the elements of S . (iv) If x 1 , x N e R ; (x.^  x N) R = (2) We shall write I <J R for "I is an ideal of R . " (3) Classes of rings will usually be denoted by script letters and all classes of rings are assumed to be non-empty. ( 4 ) If ft and H are two classes of rings we shall write W <_ ft for " M is contained in ft". (5 ) Two classes ft and are unrelated if neither ft < 11 nor W < ft . Let ft be a class of rings. We list several - 5 -c o n d i t i o n s which ft may s a t i s f y : (A) I f R e ft and R' i s a homomorphic image of R then R' e ft . (B) For any r i n g R there e x i s t s ft(R) <3 R such t h a t ft(R) e ft and i f J < R and J e ft then J c ft(R) . (D) I f every non-zero homomorphic image of a r i n g R cont a i n s a non-zero i d e a l i n ft , then R e f t . (E) Every non-zero i d e a l of a r i n g i n & can be homomor-p h i c a l l y mapped onto a non-zero r i n g i n ft . (F) I f every non-zero i d e a l of R can be homomorphically mapped onto a non-zero r i n g i n ft then R e f t . And i f ft s a t i s f i e s (B), i t may a l s o s a t i s f y : (C) F o r any r i n g R , ft(R/ft(R)) = ( 0 ) . 1 . 1 . 1 DEFINITION: ( i ) I f ft i s any c l a s s o f r i n g s , and I i s an i d e a l of a r i n g R such t h a t l e f t , then I i s a ft-ideal of R . ( i i ) A c l a s s of r i n g s ft Is a r a d i c a l c l a s s i f and o n l y i f ft s a t i s f i e s c o n d i t i o n s (A), (B) and (C) . ( i l l ) I f ft i s a r a d i c a l c l a s s then R i s ft semi-simple (ft s.s.) i f and o n l y i f ft(R) = ( 0 ) . A c l a s s o f r i n g s G i s a semi-simple c l a s s i f and o n l y i f G = the c l a s s of a l l U s . s . r i n g s f o r some r a d i c a l c l a s s a . ( i v ) I f Hi i s a c l a s s of r i n g s s a t i s f y i n g (£), = the - 6 -c l a s s of a l l r i n g s R which cannot be homomorphically mapped onto a non-zero r i n g in_77i • (v) A c l a s s of r i n g s ft i s h e r e d i t a r y i f whenever I < R e f t , I e f t . ( v i ) A c l a s s of r i n g s ffl i s a s p e c i a l c l a s s o f r i n g s i f and o n l y i f : (a) I f R e then R i s a prime r i n g . (b) Wi i s h e r e d i t a r y . (c) I f R e and R i s an i d e a l of a r i n g K , then K/(0:R) € WI where (0:R) = {x e K : xR = Rx = ( 0 ) ) . The above d e f i n i t i o n s and the f o l l o w i n g theorems can be found i n Rings and R a d i c a l s by N. J . D i v i n s k y [7]. 1.1.2 THEOREM: A c l a s s of r i n g s ft i s a r a d i c a l c l a s s i f and o n l y i f ft s a t i s f i e s (A) and (D). 1.1.3 THEOREM: I f ytfl i s a c l a s s of r i n g s which s a t i s f i e s (E) then IMyft i s a r a d i c a l c l a s s . When '\JLy\ft, i s a r a d i c a l c l a s s we w i l l r e f e r to as the upper r a d i c a l c l a s s determined by m . 1.1.4 THEOREM: I f W i s a r a d i c a l c l a s s then the c l a s s o f a l l - 7 -H s . s . r i n g s s a t i s f i e s c o n d i t i o n s (E) and ( F ) . Conversely, i f Yfl i s a c l a s s of r i n g s s a t i s f y i n g c o n d i t i o n s (E) and (P) then yY[ — the c l a s s of a l l s . s . r i n g s . Thus, G i s a semi-simple c l a s s i f and o n l y i f G s a t i s f i e s c o n d i t i o n s (E) and ( P ) . 1.1.5 DEFINITION: M i s a s p e c i a l r a d i c a l c l a s s i f and o n l y i f tt = 1/ty^for some s p e c i a l c l a s s of r i n g s • 1.1 .6 THEOREM: I f M i s a h e r e d i t a r y r a d i c a l c l a s s and no n i l -p o tent r i n g s are i n tt than tt i s a s p e c i a l r a d i c a l c l a s s i f and o n l y i f tt =\Jl,y^ where = the c l a s s of prime tt s .s . r i n g s . 1.1.7 THE LOWER RADICAL CONSTRUCTION. I f tt i s any c l a s s o f r i n g s , we d e f i n e : tt^ = the c l a s s of a l l r i n g s which are homomorphic images of r i n g s i n tt . And f o r any o r d i n a l number /3 >_ 2 , i f 0 i s not a l i m i t o r d i n a l : Hp - the 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 c o n t a i n s a non-zero i d e a l which i s i n S f l , . £-1 I f j3 i s a l i m i t o r d i n a l : - 8 -• H p = U ( W y : Y < 13} • L e t ¥ = UCMg : j3 i s a n o r d i n a l n u m b e r } . T h e n W i s a r a d i c a l c l a s s . We w i l l r e f e r t o . H a s t h e l o w e r  r a d i c a l c l a s s d e t e r m i n e d b y t h e c l a s s M . 1.1 .8 D E F I N I T I O N : ( i ) L e t R b e a r i n g , I a s u b r i n g o f R a n d N a p o s i t i v e i n t e g e r ; I i s a n a c c e s s i b l e s u b r i n g o f d e g r e e N o f R i f a n d o n l y , i f t h e r e e x i s t s 1^, . . . , 1^ c R s u c h t h a t I = I 1 < I 2 < . . . < I N A R . ( i i ) A s u b r i n g I o f a r i n g R i s a n a c c e s s i b l e s u b r i n g o f R i f a n d o n l y i f I i s a n a c c e s s i b l e s u b r i n g o f d e g r e e N o f R f o r some p o s i t i v e i n t e g e r N . I n o r d e r t o e s t a b l i s h t h e n o t a t i o n we l i s t t h e f o l l o w i n g r a d i c a l c l a s s e s a l l b u t t h e f i r s t o f w h i c h a r e d i s c u s s e d i n D i v i n s k y [7]• F F = t h e u p p e r r a d i c a l c l a s s d e t e r m i n e d b y t h e c l a s s o f a l l f i n i t e f i e l d s . F = t h e u p p e r r a d i c a l c l a s s d e t e r m i n e d b y t h e c l a s s o f a l l f i e l d s . 3" = t h e u p p e r r a d i c a l c l a s s d e t e r m i n e d b y t h e c l a s s o f f u l l m a t r i x r i n g s o v e r d i v i s i o n r i n g s . t h e u p p e r r a d i c a l c l a s s d e t e r m i n e d b y t h e c l a s s o f a l l s i m p l e r i n g s w i t h u n i t y . t i = t h e u p p e r r a d i c a l c l a s s d e t e r m i n e d b y t h e c l a s s o f a l l - 9 -simple n o n - t r i v i a l r i n g s . J = the upper r a d i c a l c l a s s determined by the c l a s s of a l l p r i m i t i v e r i n g s . j8 = the upper r a d i c a l c l a s s determined by the c l a s s 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 r i n g s w i t h idempotent h e a r t s . = the upper r a d i c a l c l a s s determined by the c l a s s of a l l r i n g s without proper d i v i s o r s of z e r o . = the c l a s s of a l l n i l r i n g s . £ = the c l a s s of a l l l o c a l l y n i l p o t e n t r i n g s . j3 = the lower r a d i c a l c l a s s determined by the c l a s s of a l l n i l p o t e n t r i n g s . ^ = the lower r a d i c a l c l a s s determined by the c l a s s of a l l n i l p o t e n t r i n g s N such t h a t N = "/^(R) f o r some r i n g R w i t h D.C.C. on l e f t i d e a l s , cj* = the lower r a d i c a l c l a s s determined by the c l a s s of a l l zero simple r i n g s . 1 . 2 RINGS WITHOUT NILPOTENT ELEMENTS. Our purpose i n t h i s s e c t i o n i s to e s t a b l i s h : 1 . 2 . 1 THEOREM: A r i n g R without n i l p o t e n t elements i s isomorphic to a s u b d i r e c t sum of r i n g s without proper d i v i s o r s o f z e r o . • I t w i l l be convenient to f i r s t prove: x - 10 -1.2.2 LEMMA: I f R has no n i l p o t e n t elements and 0 ^ x e R then ( i ) x r = [y e R : xy = 0} <1 R and x r = x t = (y e R : yx = 0} , ( i i ) x $ x t , ( i i i ) i f r e R and r x € x^ then r e x^ , ( i v ) the f a c t o r r i n g R/ x^ n a s n 0 n i l p o t e n t elements P r o o f : L e t 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 . I f a e R and ax = 0 then (xa) = 0 so xa = 0 . S i m i l a r i l y i f xa = 0 then ax = :0 . t h i s e s t a -b l i s h e s ( i ) . Since x 2 ^ 0 , ( i i ) i s c l e a r . I f a,b e R and ab = 0 then (bab) = 0 so bab = 0 , but then ( a b ) 2 = 0 so ab = 0 . Prom t h i s ( i i i ) and ( i v ) f o l l o w immediately. Q.E.D. To prove the theorem i t i s s u f f i c i e n t to f i n d , f o r each non-zero x e R , an i d e a l I ( x ) of R f o r which R/I(x) has no proper d i v i s o r s o f zero and x £ I( x ) . L e t Z(x) = ( I < R : x. $ I , i f r x e I then r e I , and R/I has no n i l p o t e n t elements} . By 1.2.2 x^ e Z(x) so Z(x) $ and i t i s c l e a r t h a t the union o f an ascending c h a i n i n Z(x) i s a l s o i n Z(x) . Thus we may choose, by Zorn's Lemma, I ( x ) maximal i n Z(x) . I f a e R and a £ l ( x ) l e t - 11 -J = ( y e R : a y e l ( x ) ) 2 x ( x ) • T h e n J / I ( x ) = ( a + I ( x ) ) p i n R / I ( x ) a n d b y 1.2.2(1) ( a + I ( x ) ) , = ( a + I ( x ) ) i n R / I ( x ) . S i n c e a l ( x ) , a x <£ l ( x ) s o x <t J . I f r x e J t h e n a r x e l ( x ) s o a r e l ( x ) , h e n c e r e J . F i n a l l y b y l,2.2.(iv) R / J = R / I ( x ) / J / I ( x ) h a s n o n i l p o t e n t e l e m e n t s , s o J e Z ( x ) . H e n c e J = l ( x ) s o R / l ( x ) h a s n o p r o p e r d i v i s o r s o f z e r o . Q.E.D. T h i s r e s u l t h a s a l s o b e e n p r o v e n b y A n d r u n a k i e v i c a n d R j a b u h i n [4] u s i n g a n a r g u m e n t i n v o l v i n g m - s y s t e m s . I n t e r m s o f t h e 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 [ 3 ] a n d T h i e r r i n [14] we c a n r e s t a t e 1.2.1 a s f o l l o w s : • A r i n g R i s ~Ylcr s e m i - s i m p l e i f a n d o n l y i f R h a s n o n i l -o p o t e n t e l e m e n t s . - 12 -CHAPTER I I 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 - C L A S S E S : We b e g i n w i t h t h e f o l l o w i n g _ d e f i n i t i o n s . 2 .1 .1 D E F I N I T I O N : A c l a s s o f r i n g s ft i s s t r o n g l y h e r e d i t a r y , i f a n d o n l y i f a l l s u b r i n g s o f r i n g s i n ft a r e i n ft . 2 . 1 . 2 D E F I N I T I O N : F o r a n y c a r d i n a l n u m b e r K , ( i ) A s u b r i n g S o f a r i n g R i s a K - s u b r i n g o f R i f a n d o n l y i f t h e r e i s a s e t A c S s u c h t h a t <A> = S a n d t h e c a r d i n a l i t y o f A i s £ K . A r i n g R i s a K - r i n g i f R i s a K - s u b r i n g o f R . ( i i ) F o r a n y c l a s s o f r i n g s ft , ft(K) i s t h e c l a s s o f a l l . r i n g s R s u c h t h a t e v e r y K - s u b r i n g o f R i s i n ft . ( i i i ) A c l a s s o f r i n g s 7 i s a K - c l a s s i f a n d o n l y i f t h e r e i s a c l a s s o f r i n g s ft s u c h t h a t 3 = ft(K) . Some i m m e d i a t e c o n s e q u e n c e s o f t h e s e d e f i n i t i o n s a r e t h e f o l l o w i n g : 2 . 1 . 3 P R O P O S I T I O N : L e t U a n d ft b e c l a s s e s o f r i n g s a n d K a n d T b e c a r d i n a l n u m b e r s . - 1 3 -( i ) 34(K) is. strongly hereditary and «(K) < 34(K)(T) . J i i ) I f 34 <. R then M(K) <. R(K) . ( i i i ) I f K <. T then H(r) < »(K) = 34(K)(T) . (iv) 34 i s a K-class i f and only i f 34 = 34(K) . (v) I f suodirect sums of rings i n 34 are i n 34 then subdirect sums of rings i n 34(K) are i n 34(K) . ' (vi) I f 34 s a t i s f i e s condition (A), so does 34(K) . Proof: (i ) I f S i s a subring of a r i n g R then subrings of S are subrings of R . So i f S i s a subring of R and R e 34(K) , S € W(K) ; and i n p a r t i c u l a r , a l l T-subrings of R. are i n »(K) so R e M(K)(r) . ( i i ) Assume that 34 <. R and R e »(K) . I f S i s a K-subring of R then S e 34 <_ R so S e R . Hence, R e R(K) . ( i i i ) Assume that K < r . I f R e 34(r) and S i s a K-subring of R then S i s also a T-subring of R since K <_ T . Thus S e 34 so R e 34(K) . Let R e 34(K)(T) and S be a K-subring of R . As above, S i s a T-subring of R so S e 34(K). Then S e 34 since S i s a K-ring. Therefore, M(K)(r) <_ 34(K) and hence by (i) 34(K)(r) = 34(K) . (iv) Assume that 34. i s a K-class. Then there i s a class of rings R such that 34 = ft(K) . Therefore, 34(K) = ft(K)(K) . However, by ( i i i ) R(K)(K) = R(K) =34, - 1 4 -so tt = .Jt(K) . (v) Suppose R i s a r i n g w i t h i d e a l s I : a e A such t h a t n(I : a e A} = (0) and R/I e tt(K) f o r a l l a e A . L e t S be a K-subring of R . Then S S + I a = -j— i s a K-subring of R / I a , hence e » • Moreover, fl(S n I Q : a e A} = (0) so by a our assumption on M , S e si . T h e r e f o r e , R e tt(K) . ( v i ) Assume t h a t R e tt(K) and the R' i s a homomorphic image of R . L e t S' be a K-subring of R 7 . Then there Is a K-subring S or R such t h a t S 7 i s a homomorphic image of S . ( i f S' i s generated by the cosets {x^} determined by ( x a ) l e t S be the s u b r i n g generated by (x } .) S i n c e R e tt(K) , S e t t and s i n c e » s a t i s f i e s ( A ) , S ' e M . There-f o r e , R' e »(K) . Q.E.D. In P r o p o s i t i o n 2.1-3 (v) - ( v i ) we see t h a t some c o n d i t i o n s on tt are i n h e r i t e d by tt(K) . U n f o r t u n a t e l y , i t i s not true t h a t tt(K) must be a r a d i c a l c l a s s whenever tt i s a r a d i c a l c l a s s . The next theorem shows t h a t t h i s s i t u a t i o n can not occur when K > ^ Q . - 1 5 -2.1.4 THEOREM: I f K i s a c a r d i n a l number £ JfQ a n d w l s a r a d i c a l c l a s s then tt(K) i s a r a d i c a l c l a s s . Proof: Assume t h a t K ^ H and i s a r a d i c a l c l a s s . * o F i r s t we show t h a t the c a r d i n a l i t y of K-subrings i s £ K . I f S i s a K-subring of a r i n g R there i s a s e t A c s such t h a t <A> = S and the c a r d i n a l i t y of A = T £ K . Thus the c a r d i n a l i t y of S < r £ K' . Now suppose t h a t A i s a r i n g , B <1 A and t h a t both A and A/B are i n M(K) . L e t A' be a K-subring of A . The c a r d i n a l i t y of A' n B < the c a r d i n a l i t y of A' £ K , so ' A' n B i s a K-subring of B . Since B e W(K), A' fl B e M ; and s i n c e - — g — ^ i s a K-subring of A/B e M(k), _^A_ g A — ± _ B e M ^ N o w • A / e M because U i s a r a d i c a l c l a s s . So we conclude t h a t i f A and A/B are i n Ji(K) then A e W(K) . (*) I f I and J are 34(K)-ideals of a r i n g R then 1 j J = Y1TIr € M^K) b y 2 ' 1 ' 3 ( v i ) ; a n d s o b y I + J e M(K) . I t f o l l o w s t h a t f i n i t e sums of «(K) - i d e a l s are H(K) - i d e a l s . (**) As we have a l r e a d y n o t i c e d , W(K) s a t i s f i e s con-d i t i o n (A) by 1.1.3 ( v i ) . - 1 6 -Now .we s h a l l show t h a t W(K) s a t i s f i e s c o n d i t i o n (B). L e t R be any r i n g and l e t I = the sum of a l l H(K)-i d e a l s of R . I f S i s a K-subring of I and x e S then x i s i n a f i n i t e sum of W(K) - i d e a l s of R , but by (**) t h i s sum of i d e a l s i s an tt(K) - i d e a l of R , so S = £{S n J : J i s an ' tt(K) - i d e a l of R} . Now the c a r d i n a l i -t y of S D J <_ the c a r d i n a l i t y of S £ K so S n J i s a K-s u b r i n g of J . Thus S fl J e 3i , so, s i n c e tt i s a r a d i c a l p r o p e r t y and S i s the sum of tt - I d e a l s , Sett. There-f o r e , I i s tt(K) and ( B ) i s e s t a b l i s h e d . The c l a s s 34(K) s a t i s f i e s (C) because of (*) so the proof i s complete. Q . E . D . 2 . 2 LOCAL CLASSES. T h i s s e c t i o n d e a l s w i t h g e n e r a l p r o p e r t i e s of V\0~ c l a s s e s . Henceforth, K - c l a s s e s w i l l be r e f e r r e d to as l o c a l c l a s s e s and we s h a l l w r i t e tt* f o r • A s u b r i n g S of a r i n g R i s tfQ-generated i f and o n l y i f S i s f i n i t e l y generated as a r i n g . 2 . 2 . 1 PROPOSITION: L e t tt be a c l a s s of r i n g s . tt* i s a r a d i c a l c l a s s i f and o n l y i f tt* s a t i s f i e s c o n d i t i o n (A) and f o r a l l r i n g s R , i f I O R such t h a t R/I e tt* and I e tt* then Rett*. - 17 -Proof: One way Is obvious s i n c e the two c o n d i t i o n s h o l d f o r a l l r a d i c a l c l a s s e s . Conversely, assume t h a t the two c o n d i t i o n s are s a t i s f i e d by ft* . L e t R be a r i n g and l e t I be the sum of a l l ' ft*-ideals of R . I f S i s a f i n i t e l y generated s u b r i n g of I then S i s a s u b r i n g of a f i n i t e sum of ft*-i d e a l s . J u s t as i n 2.1.4 our c o n d i t i o n s on ft* imply t h a t f i n i t e sums of ft*-ideals are ft*-ideals. Hence S e ft , so l e f t * . T h e r e f o r e , ft* s a t i s f i e s c o n d i t i o n ( B ) . Again as i n 2.1.4, ft* s a t i s f i e s c o n d i t i o n ( C ) ; so ft* i s a r a d i c a l c l a s s . Q.E.D. The f o l l o w i n g theorem p r o v i d e s a s u f f i c i e n t c o n d i -t i o n f o r c o n c l u d i n g t h a t c e r t a i n c l a s s e s are r a d i c a l c l a s s e s . T h i s c o n d i t i o n and the one giv e n i n 2.2.7 w i l l be u s e f u l when we c o n s i d e r s p e c i f i c l o c a l c l a s s e s i n Chapter V. 2.2.2 THEOREM: I f ft i s a c l a s s of r i n g s s a t i s f y i n g : ( i ) ft s a t i s f i e s c o n d i t i o n (A) . ( i i ) I f A . i s a r i n g , B <3 A and both B and A/B are i n ft then A e ft . ( i i i ) ft* < ft . Then ft* i s a r a d i c a l c l a s s . - 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 s a t i s f i e d . S i n c e ft s a t i s f i e s (A), by 2 . 1 . 3 ( v i ) , ft* s a t i s f i e s (A). Suppose t h a t A i s a r i n g and B <J A such t h a t A/B e ft* and B e ft* . L e t A' • be a f i n i t e l y generated ' s u b r i n g of A . Now -—g-^ s - ? A _ - E - M s i n c e A/B e ft* • Sinc e B e ft* and ft* i s s t r o n g l y h e r e d i t a r y , A 7 (1 B e ft* so by ( i i i ) A ' 0 B e ft . Th e r e f o r e by ( i i ) A' e ft so A e ft* . Q . E . D . 2 . 2 . 3 COROLLARY; In the theorem, c o n d i t i o n ( i i i ) can be r e p l a c e d by the c o n d i t i o n t h a t the union of a countable i n c r e a s i n g sequence of f i n i t e l y generated r i n g s i n ft* i s i n ft* . Proof: I n the pr o o f of 2 . 2 . 2 c o n d i t i o n ( i i i ) was needed to i n s u r e t h a t A' n B e ft . Since A' n B c A' and A' i countable i t i s c l e a r t h a t A 7 PI B i s the union of a count-ab l e i n c r e a s i n g sequence of f i n i t e l y generated s u b r i n g s . These subrings are ft* 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 . Thus the pr o o f of the c o r o l l a r y f o l l o w s immediately. Q . E . D . - 19 -2 . 2 . 4 COROLLARY: I f 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 ( i i i ) i s e q u i v a l e n t to the c o n d i t i o n t h a t f o r a l l r i n g s A , i f (0) j£ A e tt* then tt(A) / (0) . Proof: C l e a r l y c o n d i t i o n ( i i i ) Implies t h a t i f A i s a r i n g and (0) ^ A e tt* then W(A) = A ^ (0) . Conversely, assume t h a t f o r a l l non-zero r i n g s A, i f A e tt* then tt(A) ^ (0) . L e t A e tt* . Then A/H(A) e M* s i n c e tt* s a t i s f i e s c o n d i t i o n (A). Thus A/tt(A) must be (0) or e l s e i t would have a non-zero M-i d e a l . T h e r e f o r e A e tt . Q.E.D. I f we assume c o n d i t i o n s ( i ) and ( i i ) of 2 . 2 . 2 the problem of showing t h a t tt* i s a r a d i c a l c l a s s reduces to showing t h a t i f A' i s a f i n i t e l y generated r i n g and B ' O A' such t h a t A'/B' e tt* and B' e tt* then B' e M . In 2 . 2 . 2 - 2 . 2 . 4 we have accomplished t h i s by r e q u i r i n g t h a t tt*.<_ tt . Another p o s s i b i l i t y i s to show t h a t B' must be f i n i t e l y generated as a r i n g (and hence i n tt). We now t u r n 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 c o n t a i n - 20 -f i n i t e set of elements , x,T) such that f o r a i l N-N y e R y = E a . x . where the a. are i n t e g e r s depending i = l 1 1 on y . The proof of the f o l l o w i n g lemma i s based on a proof given by Jacobson (page 241)[11]. 2.2.6 LEMMA: I f A i s a f i n i t e l y generated r i n g and B « 3 A such that A/B e . then B i s a f i n i t e l y generated sub-r i n g of A . Proof: .Choose x^, • .., x^T i n A such that f o r a l l y e A/B there are i n t e g e r s a^, .. . f o r which y = Ci-. X-, + .. . + ct^x^ and such that { } contains a set of generators of A . S e l e c t i n t e g e r s Y±jYi and elements N _ b. . e B such that x.x. - T, y. ,.r x T, + b. . . Let B be i j i J K = 1 i j K K i j the s ubring of B generated by the f i n i t e set Y = fb. ., x^b. b. .x T, x^b. .xT } where a l l the s u b s c r i p t s vary I J J\ I j I J L is. I J L from 1 to N . F i r s t we s h a l l show that i f b e B then x M b e B f o r M = 1, N . C l e a r l y i t s u f f i c e s to consider the cases i n which b i s a generator of B . I f b = b. .xT or b. .. the r e s u l t - i s obvious. Suppose b = x b. . i j L 1 j Then - 2 1 -N _ e B . And I f XM b = ( x M X L ) b i j = ^ f^M^K X K + b M L ) b l j N b = X L b i j X H t h e n XM b - ( x M X L ) ( ^ i J X H ) = ( Kf x YM,L,K X K + bML' (b l j.x H) e B . S i m i l a r i l y , i f b e B then b x M € B f o r M = 1 , ..., N . Next we s h a l l show t h a t i f a e A there e x i s t N _ i n t e g e r s n. such t h a t a = E n.x. + b where b € B . 1 1=1 1 1 S i n c e (x^, . .., x N ) con t a i n s a s e t of generators o f A i t i s s u f f i c i e n t to c o n s i d e r the case when a = x. ... x. 1 X K 0 — By our d e f i n i t i o n of B i t i s c l e a r t h a t t h i s i s true when K <_ 2 . L e t M > 2 and suppose t h a t the r e s u l t holds f o r a l l K < M . Now, i f a = x. ... x. = (x. ... x. )x. ^ - l ^ x i ^ - 1 XM N _ N ( E n.x. + b)x. where b e B , then""a = E n.x.x. + j = l l l i M J s = 1 j j i M N N N bx. = E E n . y ^ K xK + E b .. + bx. and t h i s i s of M j = l K=l J ^ M * j = l J 1M XM the r e q u i r e d form s i n c e bx. e B . N Now c o n s i d e r C = ( E n.x. which are i n B : n. 1 = 1 1 1 1 are i n t e g e r s ] . C i s a subgroup of the f i n i t e l y generated a d d i t i v e group A . Si n c e C i s a subgroup of a f i n i t e l y generated a b e l i a n group, C i s f i n i t e l y generated as an a d d i t i v e group. L e t ( c ^ , c^} be a s e t of generators f o r 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 of A . Since X c B , <X> c B . Suppose N " _ z e B . Then z e A so z = £ n.x. + b where b e B c <X> 1=1 1 1 and the n^ are i n t e g e r s ; hence z - b e C . That i s , N _ S n.x. 6 C c <X> and b e B c <X> . T h e r e f o r e , B e <X> 1=1 1 1 so B = <X> . Q.E.D. 2 . 2 . 7 THEOREM: I f H i s a c l a s s of r i n g s such t h a t : ( i ) ft s a t i s f i e s c o n d i t i o n (A), ( i i ) F o r a l l r i n g s A , i f B <3 A such t h a t both A/B and B are i n ft then A e ft . ( i i i ) I f A i s a f i n i t e l y generated r i n g and A e ft then A e 3. £>. Then ft* i s a r a d i c a l c l a s s . 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 s a t i s f i e d . By 2 . 1 . 3 ( v i ) , ft* s a t i s f i e s c o n d i t i o n (A). Suppose t h a t B <£3 A and both A/B and B are i n ft* . L e t A' be a f i n i t e l y generated s u b r i n g of A . Then - — g - ^ - = A i ^ g e ft and by ( i i i ) &'^'r) g e ^ - S - Lemma 2 . 2 . 6 i m p l i e s t h a t A' fl B i s f i n i t e l y generated as a s u b r i n g - 23 -of B , so since B e 34* , A' n B e 34 . Now by ( i i ) A' e « so A i s 34* . This completes the proof. Q.E.D. 2.2.8 COROLLARY: I f 34 i s a r a d i c a l c l a s s and 34* <_ 3 . B. then 34* i s a r a d i c a l c l a s s . The next theorem provides s u f f i c i e n t c o n d i t i o n s f o r (U^)* to be a r a d i c a l c l a s s . 2 . 2 . 9 THEOREM: I f 7$, i s a c l a s s of 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) and Tfi a l s o s a t i s f i e s the c o n d i t i o n that i f R i s a f i n i t e l y 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 In 77? , then (*W»y^ ')* i s a r a d i c a l c l a s s . Proof: C l e a r l y (H^)* s a t i s f i e s c o n d i t i o n (A). Suppose that B i s an i d e a l of a r i n g A and that both A and A/B are i n (U M)* . Let A' be a f i n i t e l y generated subring of A . I f A' k l / . - ^ then there i s an i d e a l I of A' such t h a t A'/I / ( 0 ) and A'/I e YA . We s h a l l consider two p o s s i b l e cases and show that they both lead to a c o n t r a d i c t i o n . - 24 -Case 1: A' D B + I = A' . In t h i s case A / I = j = ^ B n j • b m c e KU.^) i s s t r o n g l y h e r e d i t a r y and A e (U.^)* , A' n B e C&jvJ* • Then s i n c e CU-**)* s a t i s f i e s (A), the f a c t o r r i n g A ' ^ n B o I e • T h u s A ' / 1 € ^ > * s i n c e A / / I i s f i n i t e l y generated. T h i s i s a c o n t r a d i c t i o n . Case 2: A' fl B + I / A' . A ' 4- B A ' In t h i s case g-— = —p—— c a n b e homomorphically mapped A' A' onto the non-zero r i n g A ; fl B + I ' S I N C E A ' D B + I i s a l s o a homomorphic image of A'/I e ; by assumption, A' — — ^ -g1 -'j. can be homomorphically mapped onto a non-zero r i n g i n 7&€ . But t h i s i s a c o n t r a d i c t i o n s i n c e ^ ^ ^ e ] / ^ . Since both cases l e a d to a c o n t r a d i c t i o n we conclude t h a t A' e Uyt • T h e r e f o r e , A € and so i s a r a d i c a l c l a s s by 2.2.1. Q . E . D . 2.2.10 COROLLARY: I f YVL s a t i s f i e s c o n d i t i o n ( E ) and (A) then ( U ^ ) * i s a r a d i c a l c l a s s . In p a r t i c u l a r , i f yvt i s a c l a s s of simple r i n g s then (tC^)* i s a r a d i c a l c l a s s . - 2 5 -2.2.11 COROLLARY: I f ' 34 i s a r a d i c a l c l a s s and no non-zero homomor-p h i c image of an 34 s.s. r i n g i s i n 14 then 34* = (U-u 0 0 )* i s a r a d i c a l c l a s s . The next lemma w i l l be u s e f u l i n showing t h a t 2.1.4 i s not true when K = ^ Q . R e c a l l t h a t /3 , the Baer lower r a d i c a l c l a s s , i s the lower r a d i c a l c l a s s determined by the c l a s s of a l l n i l p o t e n t r i n g s . 2.2.12 LEMMA: For-any r i n g R , i f ]3(R) = (0) then, ( i ) i f (0) / I i s an a c c e s s i b l e s u b r i n g of R then there i s an i d e a l J of R such that (0) / J c I and I® c J f o r some p o s i t i v e i n t e g e r N . ( i i ) i f (0) / I i s an a c c e s s i b l e s u b r i n g of R which i s f i n i t e l y generated as a s u b r i n g of R then there e x i s t s a non-zero i d e a l J of R which i s f i n i t e l y generated as a s u b r i n g of R and such t h a t J _ I . Proof: ( i ) We w i l l prove t h i s r e s u l t by i n d u c t i o n on N = the degree of I ' . I f N = 1 , I i t s e l f i s an i d e a l of R . Suppose t h a t the r e s u l t holds f o r a l l s u b r i n g s of degree l e s s than N . L e t I = I 1 <J I 2 <3 I ^ < • • • O I N <1 R . Now ( ( I ) j ) c I . by Andrunakievic's Lemma - 26 -(page 1 0 9 , [ 7 ] ) - Moreover, s i n c e j3(R) = (0) and ( ( I ) T ) i s an a c c e s s i b l e s u b r i n g of R i t f o l l o w s from 1 3 Lemma 33 i n D i v i n s k y [7] t h a t ( ( I ) T ) 3 4 ( 0 ) . Thus, 1 3 by our i n d u c t i o n h y p o t h e s i s , there e x i s t s (0) 4 J < R, J £ ( ( I ) ! ) 3 and ( ( ( l ) j ) 3 ) £ J f o r some i n t e g e r N . Then J c I and I 3 ^ c J so we are done. ( i i ) ' Assume t h a t (0) / I i s an a c c e s s i b l e s u b r i n g of R and t h a t I i s f i n i t e l y generated as a s u b r i n g of R . Then by ( i ) there e x i s t s (0) / J « 3 R such t h a t J c I and I N c J f o r some i n t e g e r N . L e t {z^, z^ -} be a s e t of generators of I . Then D = (z. ,. . . z . : L <_ N-1) i s a f i n i t e s e t such t h a t 1 1 L i f w e I / J then there are i n t e g e r s o^ such t h a t w = Efo^d : d e D} . Therefore I / J e 3. B . so by 2 . 2 . 6 J i s a f i n i t e l y generated s u b r i n g of R . Th i s completes the p r o o f . Q.E.D. We now presen t an example of a r a d i c a l c l a s s ft such t h a t ft* i s not a r a d i c a l c l a s s . 2 . 2 . 1 3 EXAMPLE: Le t K be the f r e e r i n g generated by two non-commuting indeterminants x and y . Def i n e : M = ( 2 y ) K - 2 7 -A B LEMMA A: I f I eJH , then I i s commutative i f and o n l y i f I c B . Proof: F i r s t we show t h a t B i s commutative. Since 2 y x e M and 2 x y e M , y - 2 x = 2 y x = 0 i n A and N _ . 2 x-y = 0 i n A . Hence, i f b e B , b = E a. x 1 i = l 1 where the a^ are even i n t e g e r s depending on b ( n o t i c e t h a t such a r e p r e s e n t a t i o n of b i s u n i q u e ) . C l e a r l y then, B i s commutative. To show the converse we b e g i n with the f o l l o w -i n g c a l c u l a t i o n s , the purpose of which i s to show t h a t i f u £ B then xu / ux or yu ^ uy . Suppose 0 4 u € A and u £ B . Then u = c^m^ + ... + a%i\ + b where b e B , the are odd i n t e g e r s and the nu are d i s t i n c t monomials (by a monomial we mean a product of x and y w i t h c o e f f i c i e n t 1 ). Moreover, we may assume t h a t a l l the = 1 . S i n c e xb - bx = 0 and yh = 0 = by we have t h a t xu - ux = x(m^ + ... + m^) - (m^ + ... + m^)x and yu - uy = y(m, + . . . + m r r ) - (nu + ... + m T r)y . I f u = x or = K/M = (2x) A = {I : I i s an a c c e s s i b l e s u b r i n g of. A}. - 2 8 -y or x + y then s i n c e xy 4 yx we are f i n i s h e d . Otherwise we may assume (by r e a r r a n g i n g the mi i f necessary) t h a t e Kx or e Ky . Suppose e Ky . Now i f xu - ux = 0 we have xm-^  + .. . + xm^ = m^x + ... + m^ -x + A where A e M . (*) Since A e M = ( 2 y ) K , we can w r i t e A = E b 1 n i where the n^ are d i s t i n c t monomials i n K and the b^ are even i n t e g e r s . Now, xra 1 + xm^ . 4 0 f o r i f so, m l + m j = 0 w h i c h i s n o t p e r m i t t e d . I f xm^ = m^x then m^ € Kx which c o n t r a d i c t s our assumption s i n c e K i s f r e e . But a g a i n , s i n c e K i s f r e e , xm^ must be equal to one of the monomials on the r i g h t hand s i d e of (*•). Thus xm, = n. f o r some j . However, s i n c e an even number of the monomials n . appear on the r i g h t hand s i d e of (*), and s i n c e n. 4 n. f o r i 4 j we must have t h a t n. = xm. f o r i / 1 or n. = m.x some i . Both of these s i t u a t i o n s l e a d us to one of the cases a l r e a d y c o n s i d e r e d , both of which l e a d to a con-t r a d i c t i o n . T h e r e f o r e , xu - ux / 0 . I f we assume m^ e Kx , we s i m i l a r i l y show t h a t yu - uy 4 0 . Thus i n any case e i t h e r xu - ux / 0 or yu - uy 0 . Now suppose I i s an a c c e s s i b l e s u b r i n g of A and t h a t I i s commutative'. I f I cf: B then by the above c a l c u l a t i o n s there e x i s t s a u e I , u £ B , such •' t h a t xu - ux 4 0 or yu - uy 5/ 0 . Now yu e l f o r - 29 -some p o s i t i v e Integer N so yu -u - u-yu = (yu - u y ) u N = 0 . Thus (yu - u y ) u N e 2K . . Since u | B , u | 2K ; t h e r e f o r e , u N £ 2K so yu - uy e 2K. Sinc e yu - uy i s a l s o i n (y)j^ i t f o l l o w s t h a t yu - uy € ( 2 y ) K = M . Since yu - uy = 0 we must have xu - ux ^ 0 . As above i t f o l l o w s t h a t xu -. ux e 2K . However, s i n c e x commutes with any monomial i n u i n v o l v e s o n l y x , xu - ux e ( y ) K • We conclude t h a t xu - ux = 0 . T h i s i s a c o n t r a d i c t i o n so I £ B . Q.E.D. LEMMA B: No non-zero a c c e s s i b l e s u b r i n g of A which i s cont a i n e d i n B i s f i n i t e l y generated as a s u b r i n g of A . Proof: By Lemma 2 . 2 . 1 2 ( i i ) i t i s s u f f i c i e n t to con-s i d e r the case when I < A and I c B . Suppose I O A , I c B and I = <b 1, b k> / (0) . We have seen t h a t the b i must be polynomials i n x with even c o e f f i c i e n t s . L e t L = maxfdegree b^ : i = l,...,k} and suppose the degree of b H = L . Now, = a^x + ... + a-jOc^ where the a^ are even i n t e g e r s . L e t w = max{n : 2 n d i v i d e s a i a l l i = 1, , L}. wL Defi n e b = x b„ e I . The s m a l l e s t power of x - 30 -which appears w i t h a non-zero c o e f f i c i e n t i n b i s wL + 1 . Since b e <b^, . b K > , b must be a sum of products of the b^'s . Because the degree of b^ <_ L f o r a l l i , each product which c o n t r i b u t e s to a non-zero c o e f f i c i e n t of b must c o n t a i n a t l e a s t w+1 w + 1 terms and hence 2 must d i v i d e a l l the co-e f f i c i e n t s of b . Th i s i s a c o n t r a d i c t i o n because the c o e f f i c i e n t s of b are e x a c t l y the c o e f f i c i e n t s of b H . Hence I cannot be f i n i t e l y generated. Q.E.D. LEMMA C: No a c c e s s i b l e non-zero s u b r i n g of A contained i n the i d e a l ( y ) A i s f i n i t e l y generated as a s u b r i n g of A . Proof: We can a p p l y Lemma 2 . 2 . 1 2 ( i i ) a g a i n so we need o n l y c o n s i d e r i d e a l s of A . Suppose t h a t 'I<3 A , (0) 4 I c ty) and I = <z.., .... z,,> where each z. 4 0 and z. = m.-. + ... + m.v f o r non-zero monomials m. . . Con-l l i K ± i j s i d e r a l l the m^ . and l e t d be the l a r g e s t i n t e g e r h such t h a t some m.^  e x h K . I t f o l l o w s t h a t i f - 31 -w e <z^, . . . z ^ > then max{h : w e x hK} <_ d . But i f w e I and w / 0 then x^ + 1w e I and x d + 1 w 4 0 . T h i s i s a c o n t r a d i c t i o n so I cannot be f i n i t e l y generated as a s u b r i n g of A . Q.E.D. We are now ready to prove: THEOREM: IX-M. i s a r a d i c a l c l a s s but (Vb^)* i s not a Pi r a d i c a l c l a s s . P r o o f : The c l a s s ^ s a t i s f i e s p r o p e r t y (E)'so }JL^A i s a r a d i c a l c l a s s and R e IXy^  i f and o n l y i f R can not be homomorphically mapped onto a non-zero r i n g i n "ftt . In order t o show t h a t i s not a r a d i -c a l c l a s s we w i l l show t h a t A/B e (]LM)* , B e (U*)* but A { (UJ* . S i n c e A i s f i n i t e l y generated and A € TVl i t i s c l e a r t h a t A £ (Um)* . To see t h a t A/B e ( t i ^ ) * , suppose that , R' i s a f i n i t e l y generated s u b r i n g of A/B . I f R' ]yC^ then R' can be homomorphically mapped onto R'/L =" I €*)tyt and I ^ (0) . Since A/B i s a r i n g of - 3 2 -c h a r a c t e r i s t i c 2 , I must be of c h a r a c t e r i s t i c 2 . C l e a r l y ( y ) A c o n t a i n s a l l a c c e s s i b l e subrings of A which are of c h a r a c t e r i s t i c 2 so I c ( y ) A • Since I i s a homomorphic image of R 7 , I must be f i n i t e l y generated as a r i n g . T h i s c o n t r a d i c t s Lemma C. There-f o r e / R' e \ so A/Be . ( U ^ * -To see t h a t B e (14*^ )* , suppose R' i s a f i n i t e l y generated s u b r i n g of B . Then R 7 i s commu-t a t i v e so i f R 7 i t can be homomorphically map-ped onto a non-zero I e t/i and I must be commutative and f i n i t e l y generated as a r i n g . By Lemma A , I c B and by Lemma B t h i s i m p l i e s t h a t I i s not f i n i t e l y generated as a r i n g . T h i s i s a c o n t r a d i c t i o n , so R 7 e U K • Hence B e . T h i s completes the p r o o f . Q.E.D. Noti c e t h a t s i n c e A i s generated by o n l y two elements, A £ ]/i (K) f o r any c a r d i n a l K such t h a t 2 £ K < H 0 * T h u s > altho u g h l l ^ i s a r a d i c a l c l a s s , 1/ (K) i s not 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 K such t h a t 2 £ K <_ ^  • . 2 . 3 ELEMENTARY CLASSES. The problems i n v o l v e d In d e a l i n g with c l a s s e s J*(K) where 2 < K < ^£ are s i m i l a r to those i n v o l v e d i n d e a l i n g - 33 -with l o c a l classes. For this reason we w i l l hot deal s p e c i -f i c a l l y with such K-classes but w i l l pass on to the proper-tie s of 2-classes. We s h a l l write tt' f o r tt(2) and r e f e r to 2-classes as elementary classes. A subring i s a 2-subring i f and only i f i t i s generated, as a r i n g , by one element. Such rings are a l l homomorphic images of the r i n g $[X] of a l l polynomials over the integers i n one variable X and with zero constant c o e f f i c i e n t . Hence a l l rings generated by one element are of the form ®X]/I where I = ( f ] L ( X ) , f R ( X ) ) f o r some f i n i t e set of elements ( f 1 ( X ) , f R ( X ) ) £ (JtX] . The proofs of the following three r e s u l t s are s i m i l a r to the" proofs of 2 . 2 . 1 , 2 . 2 . 2 and 2 . 2 . 9 respectively. 2 . 3 . 1 PROPOSITION: Let tt be a class of rings. tt-' i s a r a d i c a l class i f and only i f tt' s a t i s f i e s condition (A) and f o r a l l rings R , i f K R such that R/I € tt' and I e tt' then R e t t ' . 2 . 3 . 2 THEOREM: I f tt i s a. class of rings s a t i s f y i n g : (i ) tt s a t i s f i e s condition (A), ( i i ) I f A i s a r i n g , B <Q A and both B and A/B are i n tt then A e tt . ( i i i ) tt' < K . - 34 -Then 34' i s a r a d i c a l c l a s s . 2.3-3 THEOREM: Let 7A be a c l a s s of r i n g s s a t i s f y i n g c o n d i t i o n ( E ) . I f a l l non-zero homomorphic images of r i n g s i n YYl which are generated by one element can be homomorphically mapped onto non-zero r i n g s i n "YA , then ( t l ^ ) ' i s a r a d i c a l c l a s s . In [13] Rjabuhin d i s c u s s e s elementary r a d i c a l c l a s s e s . I f A i s a r i n g , I o A and a e A d e f i n e (I*a) = ( f ( X ) e <?(X] : f ( a ) € 1} . L e t R be any s e t of i d e a l s of fJ[X] s a t i s f y i n g : ( i ) i f A o B e R and A < f$X] then A e R . ( i i ) i f B e R and f ( X ) e ' (p[X] then (B*f) e R . ( i i i ) i f A 3 B , A e R and f o r a l l f ( X ) e A , (B*f) e R , then B e R . Rjabuhin c a l l s such a s e t o f i d e a l s an r - s e t ; and d e f i n e s f o r any r - s e t R , the c l a s s of r i n g s ft(R) to be a l l r i n g s R such t h a t ( ( ( 0 ) * a ) : a e R} £ R . He proves t h a t i f R i s an r - s e t then 5t(R) i s an elementary r a d i c a l c l a s s ( R j a -buhin c a l l s such c l a s s e s 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 ) and t h a t i f 34 i s any elementary r a d i c a l c l a s s then there i s an r - s e t R such t h a t 34 = 34(R) . We s h a l l give an out-l i n e of the p r o o f . Suppose t h a t R i s an r - s e t , then ( i ) i m p l i e s - 35 -t h a t tt(ft) has p r o p e r t y (A). Suppose B O A and both A/B and B are i n tt(ft) . Let a e A , then ((0)*a) £ (B*a) = ((0)*a) € ft where a" = a + B i n A/B . Now ( i i i ) i m p l i e s t h a t ((0)*a) e ft f o r i f f ( X ) e ((0)*a) , then ( ( ( 0 ) * a ) * f ) = ( ( 0 ) * f ( a ) ) e ft s i n c e f ( a ) e B . Thus A e tt(ft) . By the very d e f i n i t i o n of tt(ft) , tt(ft) i s an elementary c l a s s so by our P r o p o s i t i o n 2 . 3.1, tt(ft) i s a r a d i c a l c l a s s . N o t i c e t h a t ( i i ) i m p l i e s t h a t i f l e f t then P[X]/1 e tt(ft) . Conversely, i f tt' i s a r a d i c a l c l a s s l e t jaf = (I <J (p[X] .: P[X]/I e tt'3 . Since tt' s a t i s f i e s (A) ,J s a t i s f i e s ( i ) ; and 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 , <af s a -t i s f i e s ( i i ) . Suppose A 3 B and A e dl and f o r a l l f ( X ) e A , (B*f) e ft . T h i s i m p l i e s t h a t e tt' and A/B e tt' so s i n c e tt' i s a r a d i c a l p r o p e r t y , (p['X3/B e tt'. Hence B e SO <_? s a t i s f i e s ( i i i ) . T h e r e f o r e (J i s an r - s e t and c l e a r l y tt' = tt(<J ) . 2 . 3 - 4 THEOREM: L e t tt' be an elementary c l a s s which i s a r a d i c a l c l a s s and which co n t a i n s the Baer lower r a d i c a l /3 . A r i n g R i s tt' s . s . i f and o n l y i f R i s isomorphic to a sub-d i r e c t sum of prime tt' s.s. r i n g s . Hence tt' i s a s p e c i a l r a d i c a l c l a s s . - 36 -P r o o f : 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 a l w a y s s e m i - s i m p l e so we n e e d o n l y show t h a t i f R i s ft' s . s . t h e n 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 p r i m e ft's.s. r i n g s . So i t i s s u f f i c i e n t t o f i n d , f o r e a c h n o n - z e r o e l e m e n t x o f a n ft' s . s . r i n g R , a n I d e a l I o f R s u c h t h a t x I a n d R / I i s a p r i m e ft's.s. r i n g . L e t R be ft's.s. a n d 0 / x e R . Then t h e r e i s a y e ( x )R s u c h t h a t <y> W and hence <y> £ ft' . L e t Z = ( I <J R : < y ^ + I £ ft'} . Then Z / 0 s i n c e (0) e Z . Suppose : a e A i s a n a s c e n d i n g c h a i n o f i d e a l s i n Z . Then N a D <y> : a e A i s a n a s c e n d i n g c h a i n o f i d e a l s I n <y> . S i n c e <y> has A . C . C . t h e r e i s a y e A s u c h t h a t N a n <y> c n <y> f o r a l l a .e--A • . " " ' L e t N = U { N a : a e A } . Then N n < y > = N Y n < y > so • < ^ s ^ f = ^ _ <y>+N N Y ft' .. T h e r e f o r e N e Z . Thus we may c h o o s e , b y Z o r n ' s Lemma, I m a x i m a l i n Z . Since y £ I and y e ( x ) , x I . To. see t h a t R / I i s ft's.s., s u p p o s e t h a t J < R t h e n c e B u t ^ . M ' so Z M ' • Thus J / I £ ft7 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 and <y>HJ+I c ^ . T h e r e f o r e R / I i s ft's.s-. . - 37 -Next we s h a l l show t h a t R/I Is a prime r i n g . I f and Jg are I d e a l s of R which p r o p e r l y c o n t a i n I then J1 n J 2 p I . F o r i f ^ 0 ^ = 1 then < y ^ + I ' i s a <y>+J-, <y>+J 2 s u b d i r e c t sum of R 1 = — j and Rg .= — j both of whi ch are i n tt' ; hence, the ( e x t e r n a l ) d i r e c t sum of R^ and Rg i s i n tt' and 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 and the s u b d i r e c t sum i s a s u b r i n g of the d i r e c t sum, < yx+ I e tt' . Now i f J X * J 2 £ I then ( ^ n J g ) 2 c I . But J , n J • then -——: € j3 <_ tt which c o n t r a d i c t s our p r e v i o u s con-c l u s i o n t h a t R/I i s tt's.s. T h e r e f o r e , J-^-Jg £ I so R/I i s a prime r i n g . T h i s completes the p r o o f . Q . E . D . In Theorem 2.1.4 we proved t h a t i f M i s a r a d i c a l c l a s s and K i s a c a r d i n a l number such t h a t K ^ K then tt(K) i s a r a d i c a l c l a s s . In 2.2.13 we presented an example of a r a d i c a l c l a s s tt such t h a t tt(K) was not a r a d i c a l c l a s s f o r a l l c a r d i n a l s K such t h a t 2 £ K <_ . The analogous q u e s t i o n concerning elementary c l a s s e s i s unsolved. That i s , we do not know whether or not tt' = tt(2) must be a r a d i c a l c l a s s whenever tt i s a r a d i c a l c l a s s . - 38 -2 . 4 GENERALIZED K-CLASSES: In the past three s e c t i o n s we have been concerned with the q u e s t i o n , "When are K - c l a s s e s r a d i c a l c l a s s e s ? " The purpose of t h i s s e c t i o n i s to c o n s i d e r the q u e s t i o n , "When are K- c l a s s e s semi-simple c l a s s e s ? " We begin by des-c r i b i n g a c l a s s of r a d i c a l s which co n t a i n s a l l r a d i c a l s t/jj where 34 i s a semi-simple K - c l a s s . 2 . 4 . 1 DEFINITION: Let 34 be a c l a s s of r i n g s and K a c a r d i n a l number which i s >_ 2 : ( i ) 34 fir\ i s the c l a s s of a l l r i n g s R such t h a t g(K) every non-zero homomorphic image of R cont a i n s a non-zero K-subring i n 34 . ( i i ) J i s a g e n e r a l i z e d K - c l a s s i f and o n l y i f 3" = 34 ,vs f o r some c l a s s of r i n g s 34 . g(K) & I f 34 i s a semi-simple c l a s s then there i s some r a d i c a l c l a s s $$ such t h a t 34 i s the c l a s s of a l l #f semi-simple r i n g s . The r a d i c a l c l a s s <_f i s e x a c t l y the upper r a d i c a l c l a s s determined by 34 ; t h a t i s , <J = . In 2 . 3 ' 4 we s h a l l prove t h a t i f 34 i s a l s o a K - c l a s s then 1/.^  i s a g e n e r a l i z e d K - c l a s s . F o r any c l a s s of r i n g s ft and any c a r d i n a l number K >_ 2 we may form the c l a s s e s &(K) and R g ( x ) ' T h e c l a s s ft(K) i s always a K - c l a s s but need not be a r a d i c a l c l a s s - 39 -(even when ft i s i t s e l f a r a d i c a l c l a s s ) . The p r e c e d i n g t h r e e ' s e c t i o n s of t h i s chapter have been l a r g e l y concerned w i t h c o n d i t i o n s on R which imply t h a t ft(K) i s a r a d i c a l c l a s s . As we s h a l l prove below, a g ( K ) i s a l w a v s a r a d i c a l c l a s s . However the c l a s s of R g ( K ) semi-simple r i n g s need not be a K - c l a s s . Much of t h i s s e c t i o n w i l l be concerned with, c o n d i t i o n s on R which guarantee t h a t the c l a s s of R g ( K ) semi-simple r i n g s i s a K - c l a s s . In the f o l l o w i n g p r o p o s i t i o n we l i s t some b a s i c p r o p e r t i e s of g e n e r a l i z e d K - c l a s s e s and p o i n t out some r e l a -t i o n s h i p s between 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 . In 2 . 4 . 2 ( i x ) we prove t h a t i f ft i s a g e n e r a l i z e d K - c l a s s then ft(K') i s 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 K' >_ 2 . 2 . 4 . 2 PROPOSITION: Le t ft and R be c l a s s e s of r i n g s and K and r be c a r d i n a l number which are >_ 2 . ( i ) ft / T ^ N i s a r a d i c a l c l a s s . \ ' g(K) ( i i ) I f ft < R then "A„iv\ < ft„tv\ • - g(K) - g(K) ( i i i ) I f K < T then ft (irs < ft / „ v . - g(K) - g(r) ( i v ) ( wg(K) )g ( r ) ± *g(K) • (v) I f ft < R < ftg(K) then R g ( K ) = ftg(K) . ( V 1 ) . ( W g ( K ) ) g ( K ) - *g(K) l f a n d ° ^ i f *g(K) = 'g(K) f o r some c l a s s of r i n g s 3 which s a t i s f i e s ( A ) . - 40 -( v i i ) (*(r))g ( K ) = (»( r ) )g( 2 ) 1 M g ( 2 ) a n d l f * < M ( r ) > ( » ( r ) ) g ( K ) = » g ( 2 ) • ( v i i i ) (3* /x,N)(r) < (» ,„») \ y \ g ( K ) ; v - g(K ) ' g ( 2 ) - g(K) ( i x ) ( M g ( K ) ) ( r ) = ( ( M g ( K ) ) g ( 2 ) ) ( r ) l s a r a d i c a l c l a s s . P roof: ( i ) 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 w g ( K ) s a t i s f i e s c o n d i t i o n (A). We w i l l show t h a t 3i (irs a l s o s a t i s -f i e s c o n d i t i o n (D). Suppose t h a t every non-zero homomorphic image of a r i n g R c o n t a i n s a non-zero ^ g ( K ) " i d e a l . L e t R 7 be a non-zero homomorphic image of R . Then the M f v \ - i d e a l of R' con t a i n s a non-g(K) zero K-subring which i s i n M . Of course, t h i s sub-r i n g i s a l s o a K-subring of R 7 . There f o r e R e ^ g ( K ) • Now, by 1 . 1 . 2 , ^g(K) ^ s a r a d i c a l c l a s s . ( i i ) Suppose t h a t 3J <_ R and R e *g(£) • L e t R ' be a non-zero homomorphic image of R . Then R 7 contains a hon-zero K-subring which i s i n W and hence i n R . Theref o r e R e R /t,v . g ( K ) ( i i i ) Suppose that K <T and R e M g ( K ) • L e t R 7 be a non-zero homomorphic image of R . Then R 7 c o n t a i n s a non-zero K-subring which i s i n W . Since K <_ T , t h i s s u b r i n g i s a T-subring so R e Kg(r) • ( i v ) Suppose R e ( W g(K)^g ( r ) a n d l e t R ' b e a n o n ~ z e r o homomorphic image of R . Then there i s a non-zero T-su b r i n g S c R 7 such t h a t S e ^ ( K ) • Since S - 41 -i s a non-zero homomorphic image of i t s e l f S con-t a i n s a non-zero K-subring which i s i n ft . Hence R £ *g(K) ' (v) Suppose ft < ft < ftg(K) • By ( i i ) , ftg(R) < ftg(K) < <*g(K)>g(K) a n d < i V) ( J tg(K))g(K) Mg(K) :" T h u S ^g(K) — ^g(K) — ^g(K) ' A n d S O R g ( K ) = *g(K) ' ( v i ) Assume t h a t ( w g ( K ) ) g ( K ) = * g ( K ) ' T h e n s l n c e M g ( K ) s a t i s f i e s c o n d i t i o n (A) we are f i n i s h e d . Conversely, assume t h a t ft f x r s = ^„rv\ where S s a t i s f i e s c o n d i -g(K) g(K) t i o n (A). Suppose R e **g(K) a n c * R ' b e a n o n ~ zero homomorphic image of R . Then there i s a non-zero K-subring S of R' such t h a t S e JT . Since 3 s a t i s f i e s c o n d i t i o n (A) and S i s a K-subring, S 6 *g(K) = *g(K) ' b e f o r e R e ( » g ( K ) ) g ( K ) • T h e r e f o r e ftg(K) < ( f t g ( K ) ) g ( K ) s o b y ( i v ) , *g(K) = ( W g ( K ) ) g ( K ) ' ( v i i ) S i nce K >. 2 , by ( i i i ) we have t h a t ( * K r ) ) g ( K ) >. ( K ( r ) ) g ( 2 ) • Suppose t h a t R e (»(r)) ( K) a n d l e t R' be a non-zero homomorphic image of R . Then R 7 c o n t a i n s a non-zero K-subring S such t h a t S e ft(T). Hence every 2-subring of R' which i s contained i n S i s i n ft(r) n ft so c e r t a i n l y R e ( * ( r ) ) g ( 2 ) n *g(2)' I f ft < ft(r) .then by ( i i ) ftg(2) < ( « ( r ) ) g ( 2 ) - 42 -so ( « ( r ) ) g ( K ) = » g ( 2 ) . ( v i i i ) Suppose t h a t R e ( H g ( K ) ) ( r ) a n d l e f c R ' b e a n o n " zero homomorphic image of R . L e t S 7 be any 2-sub-r i n g of R 7 . Then S 7 i s a homomorphic image of 2-s u b r i n g S of . R . Since R e ( ! H g ^ ) ( r ) , S € W g ( K ) so S 7 € » g ( K ) . Th e r e f o r e R e (J» g ( K ) ) g ( 2 ) • By ( i v ) , (* g (K)-) g( 2) < W g ( K ) • ( i x ) By ( v i i i ) ( » g ( K ) ) ( r ) < (» g(K))g(2) 3 0 b y 2 ' 1 ' 3 and ( i i ) , ( » g ( K ) ) ( r ) - (» g ( K ))(r)(r) < ( ( » g ( K ) ) g ( 2 ) ) ( r ) By ( i v ) , (^ g(K)^g(2) - *g(K) s o u s i n S 2 . 1 . 3 ( i i ) a g a i n we conclude t h a t -(» g( K))(r) = ( ( M g ( K ) ) g ( 2 ) ) ( r) • Since *g(K) l s a r a d i c a ^ c l a s s , i f T ^ , then ( ^ g ( K ) ^ r ^ i s a r a d i c a l c l a s s by 2.1.4. By ( v i i i ) we know t h a t ( ^ g ( K ) ) ( r ) -^ *g(K) so when r <_ arguments e x a c t l y p a r a l l e l i n g 2.2.1 and 2.2.2 can be used to show t h a t ( ^ g ( K ) ^ ^ ^ : i s a r a d i c a l c l a s s . Q.E.D. In P r o p o s i t i o n 2.4-3 below we s h a l l prove t h a t i f G i s a semi-simple K - c l a s s then 1 ^ i s a g e n e r a l i z e d K-c l a s s . U n f o r t u n a t e l y the converse i s f a l s e s i n c e there are g e n e r a l i z e d K - c l a s s e s ^g(K) ^ o r w h i c : h ^he c l a s s of **g(K) - 43 -semi-simple r i n g s i s not a K - c l a s s . F o r example, l e t K = 2 and 34 be the c l a s s of a l l r i n g s R such that the a d d i t i v e group R + i s t o r s i o n f r e e ( t h a t i s , R e 34 i f and o n l y i f f o r a l l x e R i f hx = 0 f o r some p o s i t i v e i n t e g e r h then x = 0 ) . L e t Q = the r i n g of r a t i o n a l numbers. I f x e Q then every non-zero i d e a l I of <x> can be homomor-p h i c a l l y mapped to a non-zero 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 ( t h a t i s , a r i n g . R such t h a t nR = (0) f o r some non-zero i n t e g e r n ) . T h e r e f o r e , f o r a l l x e Q , <x> i s ^g(2) semi-simple. On the other hand Q e ** g (2) ' ^o the c l a s s of ^g(2) s e m i " s i m P l e r i n g s i s not a 2 - c l a s s . 2 . 4 . 3 PROPOSITION: Let K >_ 2 be a c a r d i n a l number. I f G i s a semi-simple c l a s s and G = G(K) then = a g ( K ) ~ Wg(K) w n e r e ft = the c l a s s of a l l r i n g s which are not i n G . 34 = the c l a s s of a l l r i n g s R such t h a t a g ( K ) ( R ) ^ (°) • Moreover, the c l a s s 34 s a t i s f i e s the f o l l o w i n g c o n d i t i o n : " I f (0) 4 T e 34 and T i s a K - r i n g then ttg(K)(T) 4 ( 0 ) . " Proof: We s h a l l b egin by showing t h a t XIQ = R g ( K ) ' Suppose t h a t R e "J^. - L e t R' be a non-zero homomorphic image of R . Then R' <| G = G(K) so there i s a K-subring T 4 (0) of R' such t h a t T $ G . Therefore - 44 -R € R g ( K ) ' Suppose R e R rTr\ • L e t . R-- be a non-zero homo-g(K) . morphic image of R . Then there i s a K-subring T of R ' such t h a t T e R . Then T $ G = G ( K ) so R' <£ G . Th e r e f o r e , no non-zero homomorphic image of R i s i n G , so R e U G • Then Ua = ^g(K) * Now we s h a l l prove t h a t ^g(K) = ^g(K) ' Suppose R e ^-g(K) * L e t R' be a non-zero homo-morphic image of R . Then there i s a K-subring T of R 7 which i s i n R , t h a t i s , T i s not 1/^s.s. Since 1XG = R g ( K ) > T i s n o t a g ( K ) s ' s * S o T e 34 . T h e r e f o r e , R £ *g(K) ' Since a g ( K ) = I A Q .> <_ ft . Thus by 1 . 4 . 2 ( i i ) , 34 / T . s < R . The r e f o r e 34 t v S = R • g(K) - g(K) g(K) g(K) To see th a t the c l a s s 34 s a t i s f i e s the d e s i r e d c o n d i t i o n , suppose t h a t (0) / T e 34 and th a t T i s a K-r i n g . Since T e 34 R g ^ ( T ) ^ (0) . So, because R'g(K) = Mg(K) ' * g ( K ) ( T ) ^ * T h i s c o m P l e t e s t h e p r o o f . Q.E.D. The c o n d i t i o n of the pr o c e e d i n g p r o p o s i t i o n w i l l be u s e f u l i n some of the f o l l o w i n g r e s u l t s . In order to make i t easy to r e f e r to t h i s c o n d i t i o n (and to another c o n d i t i o n c l o s e l y a s s o c i a t e d w i t h i t ) we make the f o l l o w i n g d e f i n i t i o n . - 4 5 -2 . 4 . 4 DEFINITION; For each c a r d i n a l number K > 2 . a c l a s s of r i n g s 34 may s a t i s f y e i t h e r of the f o l l o w i n g : C o n d i t i o n r(K) : I f (0) T e 34 and T i s a K - r i n g then » g ( K ) ( T ) 4 (0) . C o n d i t i o n s(K) : I f T i s a K - r i n g and a l l non-zero i d e a l s of T can be homomorphically mapped onto non-zero r i n g s i n 34 , then T e 34 . In order to prove a converse of P r o p o s i t i o n 2 . 4 . 3 , i t i s necessary to know something about the s t a t u s of i d e a l s which are generated by K-subrings i n *g(K) ( a n d ^ i d e a l s °f such s u b r i n g s ) . So we make the f o l l o w i n g d e f i n t i o n . 2 . 4 . 5 DEFINITION: A c a r d i n a l number K i s absorbent i f and o n l y i f whenever T i s a non-zero K-subring of R and (0) / I«3 T such t h a t I e * g ( K ) then ( l ) R e M g ( K - j , f o r a l l r i n g s R and a l l c l a s s e s of r i n g s 34 . We are mainly i n t e r e s t e d i n the s i t u a t i o n when K = 2 and when K = • In Chapter IV we s h a l l prove t h a t a l l c a r d i n a l s which are <_ are absorbent. An i n t e r e s t -i n g q u e s t i o n i s whether or not a l l " c a r d i n a l numbers are absor-bent. - 46 -2.4.6 PROPOSITION: I f K i s an absorbent c a r d i n a l >_ 2 and d i s any c l a s s of r i n g s then G <_ G(K) where G = the c l a s s of ^ g(K) s e m i ~ s - i - m P l e r i n g s . Proof: Assume R e G . L e t T be a K-subring of R . I f T £ G then ( 0 ) 4 / g ( K ) ( T ) = 1 ' S i n c e K l s a b s o r -bent e ^ g ( K ) s o R i s n o t ^ g ( K ) s * s ' T h i s i s a c o n t r a d i c t i o n / so T e G . Hence R e G(K) . . Q.E.D. 2.4.7 THEOREM: Let K be an absorbent c a r d i n a l number >_ 2 . I f G i s a semi-simple c l a s s , then: IG = the c l a s s of *g(K)' semi-simple r i n g s f o r some c l a s s of r i n g s Jt s a t i s f y i n g C o n d i t i o n r(K) . Proof: Le t K be an absorbent c a r d i n a l number _> 2 and l e t G be a semi-simple c l a s s . Assume t h a t G i s a K - c l a s s . Then by 2.4.3 G = the c l a s s of M^^^s.s. r i n g s where H s a t i s f i e s C o n d i t i o n r(K) - 47 -Conversely, assume G = the class of tt^^s.s. rings where tt s a t i s f i e s Condtion r(K) . Then by 2 . 4 . 6 G <_ G(K) . We need only show that G(K) <_ G . Let R e G(K) and l e t (0) ^  I <Q R . I f I e M g ( K ) t h e n there i s a K-subring T of I such that T e Ji . Since tt s a t i s f i e s Condition r(K) , * g ( K ) ( T ) ^ s o T £ G • T h i s i s a contradiction since we assumed that R e G(K) . Therefore I k tt f v S so R e G * g( K) • Q.E.D. 2 . 4 . 8 COROLLARY: Let K be an absorbent cardinal. I f <J i s a class rings s a t i s f y i n g condition (A) then the class of / t r x S . s . rings i s a K-class. Proof: The class p$ s a t i s f i e s Condition r(K) . In f a c t , since $ s a t i s f i e s (A) , i f T e J and T i s a K-ring then T e dg(Kj • Q.E.D. For absorbent cardinal numbers K , Theorem 2 . 4 . 7 answers the question, "When i s a semi-simple class a K-class?" The next theorem answers the question, "When i s a K-class a semi-simple class?" - 48 -2.4.9 THEOREM: Let K be an absorbent cardinal >. 2 . I f ft i s a K-class then: ft i s a semi-simple class i f and only i f ft s a t i s f i e s Condition s(K) . Proof: Let K be an absorbent cardinal >. 2 and ft a K-class. Assume ft i s a semi-simple c l a s s . Then ft s a t i s -f i e s condition (F) so c e r t a i n l y ft s a t i s f i e s Condition s(K). Conversely, assume that ft s a t i s f i e s Condition s(K) . Since ft = ft(K) , ft i s strongly hereditary so cer-t a i n l y ft s a t i s f i e s condition (E). To show that ft s a t i s -f i e s condition (F), suppose R i s a r i n g and every non-zero i d e a l of R can be homomorphically mapped onto a non-zero r i n g i n ft . I f R $ ft = ft(K) then there i s a K-subring T of R such that T £ ft . Since ft s a t i s f i e s Condition s(K), there i s a non-zero i d e a l I of T such that no non-zero homomorphic image of I i s i n ft . Because ft = ft(K) this implies that every non-zero homomorphic image of I contains a K-subring which i s not i n ft . Thus I i s ft /Tr-, where • g(K) ft = the class of a l l rings which are not i n ft . Now, since K i s absorbent, (I)T-, e ft / ^ v . But this contradicts our V 'R g(K) assumption that every non-zero i d e a l of R can be homomorphi-c a l l y m a p p e d o n t o a n o n - z e r o r i n g i n tt(K) = tt . T h e r e f o r e R e B . - T h u s , tt s a t i s f i e s b o t h c o n d i t i o n ( E ) a n d ( F ) , s o tt i s a s e m i - s i m p l e c l a s s . Q . E . D . U s u a l l y , w h e n we a r e c o n s i d e r i n g a g e n e r a l i z e d K -c l a s s tt fvs , t h e c l a s s tt s a t i s f i e s c o n d i t i o n ( A ) a n d g(K) h e n c e tt r x r s - ( t t / v N ) / „ N . I n 2 . 4 . 8 we s a w t h a t t h i s i m -g(K) v g ( K ) / g ( K ) p l i e s t h a t t h e c l a s s o f ^ g ( K ) s " s ' r i n S S i s a K - c l a s s ( p r o -v i d e d t h a t K i s a n a b s o r b e n t c a r d i n a l ) . The f o l l o w i n g d e f i n i t i o n s w i l l b e u s e f u l i n i n v e s t i g a t i n g g e n e r a l i z e d K -c l a s s e s » g ( K ) f o r w h i c h ( t t g ( K ) ) g ( K ) = ttg(K) . 2 . 4 . 1 0 D E F I N I T I O N : F o r e a c h c a r d i n a l n u m b e r K , a c l a s s o f r i n g s may s a t i s f y e i t h e r o f t h e f o l l o w i n g : C o n d i t i o n r(K) : I f (O) ^ T e tt a n d T i s a K - r i n g t h e n t h e r e i s a n o n - z e r o K - s u b r i n g L o f T s u c h t h a t L e t t f v * . C o n d i t i o n s ( K ) : I f T i s a K - r i n g a n d a l l n o n - z e r o K - s u b r i n g s o f T c a n b e h o m o m o r p h i -c a l l y m a p p e d o n t o n o n - z e r o r i n g s i n tt , t h e n T e tt . C o n d i t i o n s r ( K ) a n d s ( K ) s e e m t o b e s l i g h t l y - 50 -s t r o n g e r t h a n C o n d i t i o n s r ( K ) a n d s ( K ) r e s p e c t i v e l y . I n f a c t , i f K I s an a b s o r b e n t c a r d i n a l t h e n C o n d i t i o n r ( K ) i m p l i e s C o n d i t i o n r ( K ) . To see t h i s one need o n l y n o t i c e t h a t i n t h i s c a s e I f L i s a K - s u b r i n g o f a r i n g R and L e f t / r r - \ t h e n t h e i d e a l ( L ) D e ft /T,N . g ( K ) R g ( K ) The r e l a t i o n s h i p between C o n d i t i o n s ( K ) and s ( K ) i s n o t so c l e a r . However , i n 2 . 4.12 we a r e a b l e t o p r o v e t h a t i f K i s an a b s o r b e n t c a r d i n a l and ft <_ ft(K) t h e n ft s a t i s f i e s C o n d i t i o n s ( K ) I f ft. s a t i s f i e s C o n d i t i o n i~(K) . C o r r e s p o n d i n g t o 2 . 4.7 we p r o v e : 2 . 4.11 THEOREM: L e t K be a n a b s o r b e n t c a r d i n a l number >_ 2 . I f G i s a s e m i - s i m p l e c l a s s , t h e n : G = t h e c l a s s o f G i s a K - c l a s s a n d 'G g ( K ) i f a n d o n l y i f G ft /„\S . s . r i n g s f o r some c l a s s o f r i n g s ft s a t i s f y i n g C o n d i t i o n F ( K ) . P r o o f : L e t K be a n a b s o r b e n t c a r d i n a l > 2 and l e t G - 51 -be a semi-simple c l a s s . Assume G i s a K-cl a s s and d4.Q)g(K) ~ M»G ' Then by 2 . 4 . 3 , HG = R g ( K ) w h e r e R = t h e c l a s s of a l l r i n g s which are not i n G . Now, (R / ^ s ) f i r s •= R / v\ so g(K)'g(K) g(K) by 2 . 4 . 2 ( v i ) , R g ( K ) = ^g(K) ** o r s o m e c l a s s of r i n g s 3" s a t i s f y i n g c o n d i t i o n (A). T h e r e f o r e , 1/tQ = ^g(jc) and, s i n c e 3 s a t i s f i e s c o n d i t i o n (A), Z. c e r t a i n l y s a t i s f i e s C o n d i t i o n r(K) . Conversely, assume t h a t G = the c l a s s of 34' /,, Ns.s. g(K) r i n g s and th a t 14 s a t i s f i e s C o n d i t i o n r (K) . F i r s t we s h a l l show t h a t G i s a K - c l a s s . By P r o p o s i t i o n 2 . 4 . 6 G <f G(K) . Suppose t h a t R { G . Then M g ( K ) ( R ) ^ (°) so there i s a non-zero K-subring T of W g ( K ) ( R ) such t h a t T e 34 . Since 34 s a t i s f i e s Condition' r ( K ) there i s a K-subring L of T (and hence of R) such t h a t L e w g ( K ) • Thus, L G so R £ G(K) . So we conclude t h a t G(K) <_ G . Thus G = G(K) . Next we must show that ( I L G ^ K ) = V-Q * N O W *\JLG = M g ( K ) ' s o b y 2 , J + * 2 ( i v ) l 1 z l s s u f f i c i e n t to show t h a t *g(K) ^ ^ g ( K ) ) g ( K ) * Suppose R e S g ( K ) a n d l e t R b e a non-zero homomorphic image of R . Then there i s a non-zero K-subring T' of R' such t h a t T' e 34 . Since 34 s a t i s -f i e s C o n d i t i o n r(K) , T' cont a i n s a non-zero K - s u b r i n g L' such t h a t L' e 34 f v . . The r e f o r e R e (34 /t_. ) . ' Thi£ g(K) g(K) ;g(K) .s - 52 -completes the p r o o f . Q.E.D. Before p r o v i n g a theorem which corresponds to Theorem 2 . 4 . 9 , we n o t i c e the f o l l o w i n g r e l a t i o n s h i p between Con d i t i o n s s(K) and s~(K) . 2 . 4 . 1 2 PROPOSITION: Let K be an absorbent c a r d i n a l >_ 2 , and l e t ft be a c l a s s of r i n g s such t h a t ft <_ ft(K) . I f ft s a t i s f i e s C o n d i t i o n s(K) then ft s a t i s f i e s C o n d i t i o n s(K) . Proof: L e t K be an absorbent c a r d i n a l >_ 2 and l e t ft be a c l a s s of r i n g s such t h a t ft <_ ft(K) . Assume t h a t ft s a t i s f i e s C o n d i t i o n s(K) . L e t ft = the c l a s s of a l l r i n g s which are not i n ft . I f T i s a K - r i n g and T \ ft , then by C o n d i t i o n s(K) , there i s a non-zero K-subring L of. T such t h a t L cannot be homomor-p h i c a l l y mapped onto a r i n g i n ft . Thus L e R g ( £ ) • Now, s i n c e K i s absorbent, (L) e ft . Thus ( L ) m cannot T g(K) v 'T be homomorphically mapped onto a non-zero ft(K) r i n g . Since ft <_ ft(K) , t h i s e s t a b l i s h e s the r e s u l t - t h a t ft must s a t i s f y C o n d i t i o n s(K) . Q.E.D. - 53 -Notice t h a t , by 2.4.6, t h i s i m p l i e s t h a t i f the c l a s s of A / T,\S.s. r i n g s s a t i s f i e s C o n d i t i o n s(K) then t h i s g(K) c l a s s s a t i s f i e s C o n d i t i o n s(K) . (Provided, of course, t h a t K i s a b s o r b e n t ) . Now, corresponding to 2.4.9 we prove: 2.4.13 THEOREM: Let K be an absorbent c a r d i n a l >_ 2 . I f G i s a K - c l a s s then: G i s a semi-simple c l a s s ' G s a t i s f i e s I i f and only i f - _ and ( V G ) g ( K ) = 1AG J I C o n d i t i o n s(K) Proof: L e t K be an absorbent c a r d i n a l >_ 2 and l e t G be a K - c l a s s . Assume that G i s a semi-simple c l a s s and th a t ( U G ) G ( K ) = U Q • By 2.4.11, G = the c l a s s of ftg^s.s. r i n g s f o r some c l a s s ft which s a t i s f i e s C o n d i t i o n r(K) . Suppose t h a t T i s a K - r i n g and T k G . Then * g ( K ) ( T ) ^ (°) so, s i n c e ft s a t i s f i e s C o n d i t i o n r(K) , there i s a non-zero K-subring L of w g ( K ) ^ ) such t h a t L e ^g(^) * T n u s not a l l non-zero K-subrings of T can be homomorphically mapped to non-zero r i n g s i n G(K) = G . T h e r e f o r e , G s a t i s f i e s C o n d i t i o n s"(K) . - 5 4 Conversely, assume that G s a t i s f i e s C o n d i t i o n s(K) Then, by 2 . 4 . 1 2 , G s a t i s f i e s C o n d i t i o n s(K) , so by 2 . 4 - 9 , G i s a semi-simple c l a s s . Now, by 2 . 4 . 7 , G = the c l a s s of tt / „ v S . s . r i n g s f o r some c l a s s tt which s a t i s f i e s r(K) . g(K) & v ; Now, % a = ttg(K) and by 2 . 4 . 2 ( i v ) , ( * g ( K ) ) g ( K ) < * g ( K ) • To complete the proof we need o n l y show that ^g(K) — ^ g ( K ) ^ g ( K ) Suppose R e *g(K) a n d R ' i s a non-zero homomorphic image of R . Now, R' ^ G = G(K) . so there i s a K-subring T of R' such t h a t T $ G . Since G s a t i s f i e s C o n d i t i o n s~(K), there i s a K-subring L of . T such t h a t no non-zero homomor-ph i c image, of L i s i n G . Thus L e tt . T h e r e f o r e , g(K) R 6 ( 3 g ( K ) } g ( K ) * Q.E.D. 2.4.14 COROLLARY: Let K be an absorbent c a r d i n a l >_ 2 , and l e t tt be any c l a s s of r i n g s . There i s a c l a s s of r i n g s ft which s a t i s f i e s c o n d i t i o n (A) and such t h a t tt = the c l a s s of a g ( K ) s , s ' r i n S s l f a n d o n l y i f tt s a t i s f i e s C o n d i t i o n s"(K) and tt(K) = tt . Proof: Since any c l a s s of r i n g s which s a t i s f i e s c o n d i t i o n (A) a l s o s a t i s f i e s C o n d i t i o n r(K) , the c o r o l l a r y f o l l o w s immediately from 2.4.11 and 2.4.13. Q.E.D. 55 -We conclude t h i s chapter with a r e s u l t concerning those g e n e r a l i z e d K - c l a s s e s *g(K) f o r w h i c h l s a r a d i -c a l c l a s s . 2 . 4 . 1 5 THEOREM: Le t K be an absorbent c a r d i n a l >_ 2 . I f ft i s a r a d i c a l c l a s s and e i t h e r : ( i ) ft < ft(K) or ( i i ) i f R e f t then « g ( K ) ( R ) 4 (0) then a r i n g R i s w g ( K ) s * s * i f a n d o n l y i f R € #|( K) where M = the c l a s s of ft- s.s. r i n g s . Proof: L e t K be an absorbent c a r d i n a l > 2 and l e t be a r a d i c a l c l a s s . L e t o£ = the c l a s s of ft s.s. r i n g s , (a) Suppose t h a t ft <. ft(K) . Assume t h a t R i s a ft /„^s.s.- r i n g . I f g(K) R £ <?/(K) then there i s a non-zero K-subring T of R such that I = ft(T) 4 (0) . Since ft < ft(K) , I e ft(K) . So I (and hence, T) contains a non-zero K-subring L e f t . Since ft s a t i s f i e s c o n d i t i o n (A), L e f t /v\ and s i n c e K i s absorbent, (L)„ e ft . g(K) ' 'R g(K) Th i s c o n t r a d i c t s our assumption, so we must have that R e 5?(K) . Conversely, assume that R e ^ (K) . Then -56 -no K-subring of R i s i n ft so c l e a r l y **g(K)( R) = (°) Suppose t h a t whenever R e f t , ^ g ( K ) ^ R ) ^ ' Assume t h a t R i s . f t g ^ s . s . I f R £ J (K) then there i s a K-subring T of R such t h a t ft(T) = 1 / ( 0 ) . Then by ( i i ) , « g ( K ) ( l ) J (0) . So there i s a K-subring L of I such t h a t L e f t . J u s t as i n ( a ) , t h i s c o n t r a d i c t s our assumption. T h e r e f o r e , R e Q J ( K ) . Conversely, assume t h a t R e ^ f(K) . Then no K-subring of R i s i n ft so c l e a r l y M g ( K ) ( R ) = (°) Q.E.D. - 57 -CHAPTER I I I ELEMENTARY RADICAL CLASSES 3 . 1 THE ELEMENTARY RADICAL CLASSES & . & r . AND FC. We s h a l l b egin our study of elementary r a d i c a l c l a s s e s w i t h a d i s c u s s i o n of S and &-R . 3 . 1 . 1 DEFINITION: ( i ) Sr i s the c l a s s of a l l r i n g s R such t h a t f o r each x e R , a ^ x N + ... + a^x = 0 f o r some -integers N , • a^, ..., a^ (not a l l the a^'s are 0) which depend on ( i i ) Jo- R i s the c l a s s of a l l r i n g s R such t h a t f o r each K K—1 x e R , x + a K - 1 x + . .. + a-^ x = 0 f o r some i n t e g e r s aK-i> •••> a ] _ which depend on x . I f R € h and R' i s a homomorphic image of R then c l e a r l y R' eh . Suppose t h a t A i s a r i n g and B <J A such t h a t Bet? and A/B e tr' . L e t x e A . Then N a^x + ... + a . jX = b e B f o r some i n t e g e r s a^, . .., a^ (not a l l of which are z e r o ) . S i n c e B e ^ , there are i n t e g e r s c^, • • •, , not a l l zero, such t h a t , x, - 58 -thus £ c,( E a. •x 1) J" = E £ c .a.x11 = 0 . Th i s j = l 3 1=1 H=l j+i=H J 1 i m p l i e s t h a t A e ^ - . By the d e f i n i t i o n i t Is c l e a r t h a t Sr i s an elementary c l a s s so by 2.3«'l Ss- i s a r a d i c a l c l a s s . A s i m i l a r argument shows t h a t & R i s a l s o a r a d i c a l c l a s s . Both ir and & R are c l e a r l y elementary c l a s s e s which con-t a i n the c l a s s of a l l n i l p o t e n t r i n g s ( i n f a c t , one e a s i l y sees t h a t j8 < 71 <_ <_ ^  ) . Combining the above remarks with 2.3-4 we have: 3.1.2 PROPOSITION: Jr and js- R are s p e c i a l elementary r a d i c a l c l a s s e s . 3-1.3 LEMMA: I f I <J <?[X] and I / (0) then $[X]/I e tr . Proof: L e t (0) 4 K &[X) - and- f'(X) e @[X] . Choose g(X) / 0 , g(X) e I . L e t 0^, .. ., a K be the r o o t s of g(X) . Since the are a l l a l g e b r a i c numbers, so are = f(o.^) . Thus there are non-zero polynomials h^(X) e &[X] K such that h.(y,) = 0 . L e t h(X) = J J h.(X) , then there 1 • i = 1 1. • are i n t e g e r s a^, a M such t h a t h(X) = a^X + ... + a^X1^. Consider h(f(X)) = l(X) e Q[X] . Now l(a±) = 0 f o r a l l r o o t s a,, C L . of g(X) , so there 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 that -t(X) = d(X)-g(X) . Since the c o e f f i c i e n t s of d(X) are r a t i o n a l there i s an in t e g e r n such that n«d(X) has i n t e g e r c o e f f i c i e n t s . Now n-h(f(X)) = na ]_f(X) + ... + n a M ( f ( X ) ) M = nd(X).g(X) e I . Thus, na 1F(xy + + naM(f7xT)M = 0 i n f?[X]/I . There-f o r e , fax]/I € U • 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 of Niven and Zuckerman [12]. Lemma 3'1«3 i m p l i e s that &> i s the l a r g e s t elementary r a d i c a l c l a s s (except f o r the c l a s s of a l l r i n g s ) . I n the f o l l o w i n g p r o p o s i t i o n we c o l l e c t some i n f o r m a t i o n about the r e l a t i o n s h i p between <$3- , jfirR and the well-known r a d i c a l classes l i s t e d i n Chapter I . 3.1 .4 PROPOSITION: ( i ) I f •. tt' i s an elementary r a d i c a l c l a s s and tt' i s not the c l a s s of a l l r i n g s then tt' <_ $r . ( i i i ) N either nor i s r e l a t e d to )fl or J . Proof: ( i ) Assume that tt' i s an elementary c l a s s and tt' i s not the c l a s s of a l l r i n g s . Then fax] tt' ; f o r i f fax] e tt' then every r i n g generated by one - 60 -element (being a homomorphic image of (?[X] ) i s i n M7 , so U' i s the c l a s s of a l l r i n g s . Let R e M7. I f x e R then <x> ~ (?[X]/I e . Since 6>[X] £ «' , I / (0) so by 3 . 1 - 3 <x> e U < There-f o r e R efr , so W <_ & .' ( i i ) I f R i s a n i l r i n g and x e R then xn^x^ = 0 f o r some non-zero i n t e g e r n(x) . C e r t a i n l y then, Yl <L &R <_ $r • However, a l l f i n i t e f i e l d s are i n d£-R and &r but not i n % . Hence )t £ «(rR and ^ £ . The r i n g f?[X]/(2X) £ & R but i s i n ir so & R £ & . ( i i i ) C l e a r l y & i ftg , J R ± tfg , & i J and j £ R £ • J . For example, consider any f i n i t e f i e l d F . F e &-and' F e but F i Yl and F i J . R g r To see that J £ consider the r i n g R of a l l formal power s e r i e s i n one indeterminate x over the r a t i o n a l f i e l d Q . Let R 7 be the i d e a l of R c o n s i s t i n g of a l l i S a.x e R such that a = 0 . I t i s well-known (and easy 1=0 1 0 to prove) that R 7 e J . However, i f a^, a^ are i n t e -K gers, a^x + ... + a^ .x / 0 unless a^ = ... = a^. = 0 . Thus, R 7 j£ . This example shows that J £ J r R and Let Q(x) 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 i n an indeterminate x over the r a t i o n a l f i e l d . Q . Then - 61 -R = ( Q ( x ) ) 2 $ the r i n g of 2x2 matrices over (x) , i s i n -ft . However, R and R £ $ r , because i f a^, . .., a^ - are i n t e g e r s such that 5 fx O K ^ 0 /x 0\K fO 0\ aiVo oJ + •'• + aK\o oJ - VO o7 then a^x + ... + a^x^" = 0 i n Q(x) so a^ = ... = a^ = 0 Q.E.D. We now turn to a b r i e f c o n s i d e r a t i o n of the r a d i c a l c l a s s PC . 3.1.5 DEFINITION: . ( i ) Let p be a prime number. FC^ Is the c l a s s of a l l r i n g s R such that f o r each x e R there i s a p o s i -t i v e i n t e g e r a(x) such that p a ^ x ) . x = 0 . ( i i ) FC i s the c l a s s of a l l r i n g s R such that f o r each x e R there i s a non-zero i n t e g e r n(x) such that n(x)«x = 0 . Let p be a prime. C l e a r l y PC^ s a t i s f i e s con-d i t i o n (A). Suppose that B ^ A and both B and A/B are i n PC p . Let x e A , then p°x e B f o r some i n t e g e r a . But since p x e B e PC , P (p x) =0 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 c l e a r 3? from the d e f i n i t i o n that PC^ i s an.elementary c l a s s so by 2.3*1, FC i s an elementary r a d i c a l c l a s s . A s i m i l a r - 62 -argument shows that PC i s an elementary r a d i c a l c l a s s . '3.I.6 PROPOSITION: ( i ) FC i s an elementary r a d i c a l c l a s s , ( i i ) For each prime p , FC i s an elementary r a d i c a l c l a s s . ( i i i ) For a l l r i n g s R , PC(R) = Q (PC ( R ) : p i s a prime}. Proof: We have al r e a d y seen that the classes mentioned i n ( i ) and ( i i ) are elementary r a d i c a l c l a s s e s . To see that ( i i i ) i s t r u e , n o t i c e that R e PC i f 4. and only i f the a d d i t i v e group R i s t o r s i o n , and R e FC i f and only I f R + i s a p-group. Pa r t ( i i i ) of the propo-s i t i o n now f o l l o w s from 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" AND £>' . 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 N such that N = 77(R) f o r some r i n g R with D.C.C. on l e f t i d e a l s . This c l a s s con-t a i n s *J which i s the lower r a d i c a l c l a s s determined by the c l a s s of a l l zero simple r i n g s . In t h i s s e c t i o n we s h a l l consider the classes rj' and fa ' . We begin w i t h the f o l l o w i n g r e s u l t about the - 6 3 -c l a s s <$ . 3 - 2 . 1 PROPOSITION: J = js n P C . Proof: Let ft = the c l a s s of zero simple r i n g s . I f S e ft then the a d d i t i v e group S i s a simple a b e l i a n group; so, S + i s the c y c l i c group of p elements f o r some prime p . Therefore, ft <_ PC and so J <_ PC because J 9 Is the lower r a d i c a l c l a s s determined by ft . Since J <. P , j < jS fl F C . Suppose that jS fl PC £ J . Then there i s a r i n g R which i s i n /3 D PC but i s J semi-simple. Since R e )3 there i s a non-zero i d e a l I of R such that I = ( 0 ) . Suppose i ' i s a non-zero homomorphic image of I . Since R e PC , i ' e PC . Thus, there i s a prime p and an element x' e I ' such that x' / 0 but px' = 0 . Since ( l ' ) 2 = ( 0 ) , <x'> <j 1 7 , and <x'> i s isomorphic to the zero r i n g on the c y c l i c group of p-elements. But then <x'> e ft so I e ft^ (see 1 . 1 . 7 ) and hence I e «jf . This i s a c o n t r a d i c t i o n . Therefore j3 fl FC <. J , so = j8 fl FC . Q.E.D. 3 . 2 . 2 THEOREM: D ' = $ ' = Ti n PC - 64 -Proof: F i r s t we s h a l l show that J' = Yl n FC . Notice that j8' < yi' =Jl since P <Jl ; and that yi' < P>' , f o r i f <x> e Yl then <x> i s n i l p o t e n t so <x> e j3 . Thus Since, J = j3 n FC (by 3.2.1), J < 0 and J ° < FC . Therefore, g / ' = £' = "ft and <_ FC' = FC so j' < Yir\ FC . Suppose R e Yl D FC , and l e t x e R . Then <x> i s f i n i t e since there are p o s i t i v e i n t e g e r s K and N such that x K = 0 and Nx = 0 . Thus <x> has D.C.C. and so, since <x> e Yl , <x> e d (see Lemma 28 i n Di v i n s k y [7]) . Therefore, Yl n FC <_ so j p ' = Yl D FC . Since J < £ ) , < / ' < £ ) ' . Suppose <x> e B' but <x> <J ' . The r i n g <x> must be n i l since B' ^¥1' = H ' s 0 < x > i F C • L e t < x > = <x>/FC<x> . F i r s t we s h a l l prove that the r i n g <x>/<x>2 ^ FC . Suppose — — 2 nx e <x> . Then there are in t e g e r s a^, . .., a^ such that _ M nx = S a. x . By successive s u b s t i t u t i o n s f o r nx we see 1=2 1 that 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 j£ <x> i s n i l p o t e n t , n x = 0 f o r some i n t e g e r K , and since <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 m p l i e s that x = 0. This i s a c o n t r a d i c t i o n on since <x> ^ FC . Therefore 2 ' <x>/<x> FC so <x> can be homomorphically mapped to a zero r i n g of c h a r a c t e r i s t i c zero. Since <x> Is generated - 65 -by one element t h i s r i n g must be isomorphic to C the zero r i n g on the i n f i n i t e c y c l i c group. Since <x> e $ , C" e © . But c" <j: §b (see Theorem 14, Di v i n s k y [7]) so t h i s i s a c o n t r a d i c t i o n . Hence ^)' <_ J*' so §b' = ^f' arid the proof i s complete. Q.E.D. 3.3 CLASSES tt FOR WHICH tt' =71 . In [9] Goldman defines a H i l b e r t r i n g to be a commutative r i n g R w i t h i d e n t i t y such that J ( R ' ) = Yl(R') f o r a l l homomorphic images R' of R . He proves that i f R i s a H i l b e r t r i n g then so i s the polynomial r i n g R[X] . Since the r i n g of 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 p r o p o s i t i o n i s a s p e c i a l case of Goldman's theorem. •. 3.3.1 PROPOSITION: I f R i s a r i n g generated by one element then J(R) = YKR) • 3.3.2 THEOREM: I f j3 <_ tt < FF then tt' = 71 . Proof: Suppose that tt i s a c l a s s of r i n g s such that J3 <_ W <_ FF . - 66 -Since j8 <_ 34 , Y\ = /3' <_ 34' . Assume that FF' _^ "Yl . Then there i s a r i n g <x> e FF' such that >£(<x>) = (0) . By 3-3-1 J(<x>) = (0) and so, since <x> i s commutative, by Lemma 87 i n D i v i n s k y [7] F(<x>) = (0) . I t f o l l o w s that <x> i s a s u b d i r e c t sum of f i e l d s each of which i s generated by one element. To reach 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 to show that any f i e l d which i s generated as a r i n g by one element must be f i n i t e . Let <y> be a f i e l d and suppose <y> i s of c h a r a c t e r i s t i c zero. Then <y> contains a copy of the r a -t i o n a l s . Choose f ( y ) = a-^ y'+ ... + a^y^ of minimal degree so that N = f ( y ) i s a non-zero i n t e g e r i n <y> . Let p be a prime which does not d i v i d e any of a^, ..., a^ and l e t 1/p = c^y + ... + c n y n . By continued s u b s t i t u t i o n s f o r a^y^ 4" 1 = Ny - a-^y2 - ... - a^.^Y^ we see that f o r some p o s i -t i v e i n t e g e r L , a^/p = b^y + ... + b k y k . But then 0 = Na£ - Na£ = ( a ^ - pNb^y + ... + (a£ + 1 - p N b k ) y k . Since p does not d i v i d e any of the a i not a l l of the c o e f f i c i e n t s i n t h i s expression are 0 . Since <y> has no proper d i v i s o r s of zero i t f o l l o w s that there are i n t e g e r s d-p d^ with d 1 4 0 such that d^y + . .. + d^y = 0 and I <_ k . But- then d^ = -d^y - . .. - d^y which c o n t r a d i c t s the m i n i m a l i t y of k . Thus <y> 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 - 67 -<y> must be a l g e b r a i c i t f o l l o w s that <y> i s f i n i t e . This i s a c o n t r a d i c t i o n and so FF' <_yi . Q.E.D. This theorem may be paraphrased 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 only i f no subring of R which i s generated by one element can be homomorphically mapped onto a f i n i t e f i e l d " . 3-4 ELEMENTARY RADICAL CLASSES WHICH ARE = FC . We w i l l begin 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 of the elementary r a d i c a l c l a s s fi' . This r a d i c a l c l a s s i s u n r e l a -. ted to a l l of the well-known r a d i c a l classes l i s t e d i n Chapter 1. In f a c t , a l l S' r i n g s are H semi-simple. This r a d i c a l plays a c e n t r a l role, i n our di s c u s s i o n s concerning r a d i c a l classes which contain only T i semi-simple rings.' 3.4.1 DEFINITION: £ i s the c l a s s of a l l Idempotent r i n g s (that i s , a l l r i n g s R such that R = R ). 2 Let R be a r i n g and x e R . C l e a r l y <x> = <x> i f 2 and only i f x e <x> and hence i f and only i f there are in t e g e r s K i B. a, such that x = S a.x . Using t h i s c h a r a c t e r i -1=2 1 z a t i o n i t i s c l e a r that homomorphic images of £'-rings are In &' and tha t i f A i s a r i n g w i t h an i d e a l B. such"'that both - 68 -A/B and B are i n e' then A e t' . Therefore, by 2 . 3 - 1 , £' i s an elementary r a d i c a l c l a s s . 3-4 .2 PROPOSITION: A non-zero s'-ring without proper d i v i s o r s of zero i s an a l g e b r a i c f i e l d of prime c h a r a c t e r i s t i c . Proof: Let R be a non-zero S'-ring without proper d i v i -sors of zero. I f 0 ^  x e R then there are in t e g e r s K i K i - 1 a0, ..., av such that x = E a.x , hence ev = E a.x i - " 2 I — 2 i s an i d e n t i t y f o r <x> . Let w e R . Then x(e w - w) = (we„ - x)w = 0 , so e w = w . S i m i l a r i l y , we v = w so e„ 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 e y e <v>-v c Rv . Therefore R = Rv f o r a l l non-zero v e R ; so, since R / (0) , R i s a d i v i -s i o n r i n g . ' Let e be the I d e n t i t y of R . Then <e> f the p r i n g of i n t e g e r s s i n c e <2e> = <2e> = <4e> . Therefore the c h a r a c t e r i s t i c of R i s a prime. Since e = e e <w> f o r w a l l non-zero w e R , R i s a l g e b r a i c . Therefore, by Theorem 2 on page 183 of Jacobson [ 11] , R i s a f i e l d . Q.E.D. - 69 -3 . 4 . 3 COROLLARY: I f (0) / R e ?/ then R i s isomorphic to a sub-d i r e c t sum of a l g e b r a i c f i e l d s of p r i m e . c h a r a c t e r i s t i c . So, i n p a r t i c u l a r , R i s commutative. Proof: Let (0) R e &' and x e R . I f x N = 0 then <x> = <x> = ... = <x> = (0) so x = 0 . Hence 6 7 r i n g s 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 from 1 .2.1 and 3 - 4 . 2 . Q.E.D. Prom 2 . 1 . 3 ( i i i ) we know that e* <_ t' = ( £ ' ) * . The f o l l o w i n g theorem provides a c h a r a c t e r i z a t i o n of £' which makes i t c l e a r that i n f a c t £* = t' . 3 . 4 . 4 THEOREM: A r i n g R e &' i f and-only i f every non-zero f i n i t e l y generated subring of R i s isomorphic to a f i n i t e d i r e c t sum of f i n i t e f i e l d s . Proof: Assume that R e £' and l e t R' be a non-zero f i n i t e l y generated subr i n g of R . Then R ' e 5' so by 3 - 4 . 3 R' i s commutative. Since R 7 i s f i n i t e l y generated and commutative we may conclude, from the H i l b e r t B a s i s - 70 -Theorem, that R 7 s a t i s f i e s A.C.C. I f P' i s a prime i d e a l of R' and P' / R' then P' i s a maximal i d e a l because by 3 . 4 . 2 R'/P' i s a f i e l d . Since R' i s f i n i t e l y generated, commutative, and <g> has an i d e n t i t y f o r each generator g of R' ; R' has an i d e n t i t y ( i f , f o r i = 1, 2 , e± i s an i d e n t i t y f o r <g.^ > then e 1 + e 2 - e^e^ i s an i d e n t i t y f o r <g ,g 2>). Now-, by Theorem 2 , page 203 of Z a r i s k i and Samuel [ 1 5 ] , R' s a t i s f i e s D.C.C. Then R' i s a commutative Wedderburn r i n g so R' i s isomorphic to a f i n i t e d i r e c t sum of f i e l d s . These f i e l d s must be f i n i t e , s i n c e they are f i n i -t e l y generated and by 3 - 4 . 2 they are a l g e b r a i c of prime c h a r a c t e r i s t i c . The converse i s obvious; i n f a c t , i f x e R' and R' i s isomorphic to a f i n i t e d i r e c t sum of f i n i t e f i e l d s then there i s an i n t e g e r n(x) _> 2 such that x n ^ x ^ = x . Q.E.D. 3 . 4 . 5 COROLLARY: A r i n g R e &' i f and only I f f o r each x e R there i s an i n t e g e r n(x) >_ 2 such that x n ^ x ^ = x . 3 . 4 . 6 COROLLARY: A r i n g R e e' i f and only i f f o r a l l 0 4= x e R , <x> Is isomorphic to 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 . I f (0) / R e e' and R has D.C.C. then R i s a - 71 -commutative Wedderburn r i n g so R i s isomorphic to a f i n i t e d i r e c t sum of f i e l d s i n Z' . In the•next theorem we see that a c o n d i t i o n which i s apparently weaker than D.C.C. i s s u f f i c i e n t to ob t a i n t h i s r e s u l t . 3-4 . 7 THEOREM: I f (0) / R e Z' and 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 then R i s Isomorphic to a f i n i t e d i r e c t sum of f i e l d s i n Z' (that i s , a l g e b r a i c f i e l d s of prime c h a r a c t e r i s t i c ) . Proof: Let R be a non-zero s ' - r i n g which 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 . By 3.4 . 3 R i s commutative. Since A.C.C. on l e f t a n n i h i l a t o r s i s equivalent to D.C.C. on r i g h t a n n i h i l a t o r s , R 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 R = A^^ © . ... ffc A R $ B K + 1 where the A are f i e l d s i n z' . I f h a s n o P r o P e r d i v i s o r s of zero then, by 3-4 . 2 , B K + 1 i s a f i e l d i n z' . I f there are non-zero elements b 2 - , b 2 € "^K+l s u c n that b ] _ * b 2 = 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) = the a n n i h i l a t o r of A 1 Q .. . © a K © b^R / (0) - 72 -and A K + 1 E B K - f l ' 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 contained i n B ^ + ^ • Now, I f there are non-zero elements x,y e such that xy = 0 then D = (0 : + yR) i s a non-zero a n n i h i l a t o r (x e D ) such that D p A K - K L ( Y ^ ' ^bis c o n t r a d i c t s the m i n i m a l i -ty of • Therefore A K + ^ has no proper d i v i s o r s of zero so ^ i s a f i e l d i n &' by 3^-2. Since A ^ + i h a s an I d e n t i t y , a K+1 i s a d i r e c t summand of B K + - ^ . T h a t - i s , there i s an I d e a l B K + 2 of B K + 1 such that B K + 1 = • A K + 1 @ B K+2 ' T H E R E F O R E B K+2 ^ R a n d R = A l ® " ' ® A K + 1 © B K + 2 ' Notice that t h i s proof i s v a l i d when K = 0 , and since (0) 4 R we can begin the above process. 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 , the process above must stop. That i s , f o r some n , B n has no proper d i v i s o r s of zero and hence i s a f i e l d In £' . This completes the proof. Q . E . D . This completes our i n v e s t i g a t i o n of the elementary r a d i c a l c l a s s s' . We now present a c l a s s i f i c a t i o n of a l l elementary r a d i c a l s which are < PC . 3-4.8 DEFINITION: Define to be ft fl FC^ where Jt i s any c l a s s - 73 -of r i n g s and p i s a prime number. 3 . 4 . 9 PROPOSITION: I f 347 i s an elementary r a d i c a l c l a s s and R i s a r i n g then ( t t ' . n PC)(R) = © { » p ( R ) : p i s a prime} -Proof: Let 347 be an elementary r a d i c a l c l a s s and l e t R be a r i n g . Since i n t e r s e c t i o n s of r a d i c a l c l a s s es are r a d i c a l c l a s s e s , M p(R) and ( 34 7 n FC)(R) are def i n e d . From 3 » 1 . 6 ( i i i ) we know that f o r any r i n g A , FC(A) = © { F C ( A ) : p i s a prime} . So ( 34 7 n PC ) ( R ) = © ( F C p ( ( 347 fl PC)(R)) : p i s a prime} . Now, FC ( ( » ' fl FC)(R)) e 347 since 347 i s h e r e d i t a r y ; t h e r e f o r e , • ir FC (( 347 n. FC ) ( R ) ) c 34'(R) . Since 347 (R) c ( 347 n PC)(R) and P P P Mp(R) € FC p , Jl p(R) c P C p ( ( l i ' n FC ) ( R ) ) . Thus, J4 p(R) = F C p ( ( 347 n p c ) ( R ) ) . This completes the proof. Q.E.D. 3 . 4 . 1 0 DEFINITION: A set of p o s i t i v e i n t e g e r s i s a C.U.D. set of in t e g e r s i f and only i f whenever n e S and k i s a p o s i t i v e i n t e g e r which d i v i d e s n , k e S . Suppose that S i s a C.U.D. (closed under d i v i s o r s ) set of in t e g e r s and p i s any prime number. Then f o r each - 74 -n e S we consider the f i n i t e f i e l d F = the f i e l d of p n ^ P elements. Since S Is a C.U.D. s e t , i f n e S and k di v i d e s n then k e S • hence, the set (F• : n e S) of 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 k which P P d i v i d e s n . Let ft be the set of a l l p o s s i b l e f i n i t e d i r e c t sums of the 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 which we are d e f i n i n g i n the f i r s t p a r t of the f o l l o w i n g d e f i n i t i o n . Notice that the set of f i e l d s (F : n e S} £ ft' , P so as the C.U.D. set S changes so does R . The dependence of ft' on the prime p i s obvious. 3 -4.11 DEFINITION: ( i ) ^ ( S ) i s t h e c l a s s of a l l r i n g s R with the property i r that f o r a l l non-zero x e R , <x> i s isomorphic to a f i n i t e d i r e c t sum of 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. set of i n t e g e r s . ( i i ) 3 7l(S) i s the c l a s s of a l l r i n g s R w i t h the pro-p e r t y that R e FC p and f o r a l l x e R the f a c t o r r i n g <x>/y\(<x>) i s i n IT (S) . 3? I t i s c l e a r from the above d e f i n i t i o n that ^ ( S ) <L SpTK"3) • I n t h e f o l l o w i n g p r o p o s i t i o n we prove that these classes are r a d i c a l c l a s s e s . - 75 -3 . 4 . 1 2 PROPOSITION: I f S i s a C.U.D. set of p o s i t i v e Integers and p i s a prime number then ^p(S) and 3pT|(S) are elementary r a d i c a l c l a s s e s . Proof: Let S be a C.U.D. set of p o s i t i v e i n t e g e r s and l e t p 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 that 3p(S) i s an elementary c l a s s . I t i s a l s o c l e a r that ^p(S) s a t i s f i e s c o n d i t i o n (A)'. Suppose that B i s an i d e a l of a r i n g A and both B and A/B are i n 3 (S) . By 3 . 4 . 6 , both B and A/B are i n e' so A e &' . Let 0 / x e A . Then by 3 - 4 . 6 again, <x> i s isomorphic to a f i n i t e d i r e c t sum of f i e l d s . Therefore <x> fl B i s a d i r e c t summand of <x> so <x> = (<x>/<x> n B) Q> (<x> fl B) . Now because A/B and B are i n ^p(^) > ^ n e f i e l d s i n question must be of the form P where a e S . Therefore, A e 3 (S) ; so, by 2 - 3 . 1 , P P 3" (S) i s a r a d i c a l c l a s s . P Suppose R e ^YKS) and that R' i s a homomorphic sr image of R . Let x' e R' . Then there i s an x e R such that <x'> i s a homomorphic image of <x> . So <x'>/Yl (<x '>) i s a homomorphic image of <x>/Yl(<x>) , hence <x />/7i(<x />) e 3 (S) . Since R e PC , R' e PC . There-' p p ' p f o r e R' e Z 72(s) s o ^TKS) s a t i s f i e s c o n d i t i o n (A) . P P - 76 -Suppose that B i s an i d e a l of a r i n g A and that both A/B arid B are i n 3" % ( S ) . Both A/B and B are 3P i n PC so A e FC P P Let x e A and l e t <x> = <x>/<x> 0 B . Then <x>/yi(<x>) e 3"p(S) so <x>/Yl(<x>) i s f i n i t e . Thus, by 2 . 2 . 6 , 'Yl(<x>) i s f i n i t e l y generated as a r i n g and so; since 7l(<x>) i s a l s o n i l p o t e n t and i n PC p , Yl(<x>) i s f i n i t e . Therefore <x> must be f i n i t e . Now, by 2 . 2 . 6 again, <x> fl B i s f i n i t e l y generated as a r i n g . Since <x> fl B/Yl(<x> n B) has no non-zero n i l p o t e n t elements, 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) i s f i n i t e . Now, j u s t as above, <x> fl B i s f i n i t e . Therefore, <x> i s f i n i t e . Let N = yi(<x>) . Since <x> i s f i n i t e and commutative, <x>/N i s a f i n i t e d i r e c t sum of f i e l d s . Thus <x> f l B + N N i s a d i r e c t summand of <x>/N . Let <|> = | @ < x > n B + N M , S . N C E . N N { < X > N B ) = ^ ( < X > N B ) } <x> n B + N ^  <x> n B • m, <x> n B + N T / e v N = Yl(<x> n B) ' T H U S ' N e 3 p(S) •. IVT«,T L ~ <X> / < X > fl B + N ~ < X > M . T AT • N o w > N = I T / — N = <x> fl B + N • T H U S L ^ 1 3 A homomorphic image of <x> which has no non-zero n i l p o t e n t elements. Hence L/N i s a homomorphic image of <x>/fl(<x>) e 3-p(S) . Thus LA e y s ) . Therefore <x>/M e 3 (S) .so A e Z yi(S)' . - 77 -From the d e f i n i t i o n i t i s c l e a r that 3Tp">?(s) l s a n elementary c l a s s so by 2 . 3 . I 3" 7J(S) i s a r a d i c a l c l a s s . Q.E.D. The r a d i c a l c l a s s es 3 ( S ) , ^ p W s ) a n d p c p w i l l be our b a s i c b u i l d i n g blocks 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 which are <_ FC . We begin w i t h the fol l o w -i n g r e s u l t . 3 . 4 . 1 3 PROPOSITION: I f tt' I S an elementary r a d i c a l c l a s s and p i s a prime then nVfc = ((0)) or Yl fl FC p < ttp . Proof: Let if' be an elementary r a d i c a l c l a s s . I f & p n 71 / ( ( 0 ) } then there i s a non-zero r i n g R e H p fl Yl . Since R e FC p and R e Yi there i s an x e R , x / 0 p such that x = 0 and px = 0 .' Thus <x> = C = the zero P r i n g on the c y c l i c group of p elements, so C e tt' . P P Let A 6 71 n FC p . Suppose A ttp . Then A = A/Bp (A) / ( 0 ) . Since A <t ttp there i s an x e A such that <x> ttp . Let ( 0 ) / <x> = <x>/ttp(<x>) . Now, <x> e Yl C\ FC p so there i s a y e <x> such that py = 0 and y 2 = 0 but y / 0 . Let Y = ( y ) < ~ > • Then Y 2 = ( 0 ) and pY = 0 . So i f w e Y , <w> «? Y and <w> = C e B' • P P - 78 -Therefore Y e Mp . This i s a c o n t r a d i c t i o n . Hence A e ftp so Y l n pc p < r . Q.E.D. Suppose that ft^ : a e A i s a c o l l e c t i o n of r a d i c a l c l a s s e s such that »fi(R) n S ft (R) = (0) f o r a l l j3 e A and a l l r i n g s R . Then f o r any r i n g R we can form the d i r e c t sum (ft (R) : a e A} . In such a s i t u a t i o n we s h a l l denote by © {ft^ : a e A} the c l a s s of a l l r i n g s R f o r which R = © ( » (R) : a e A} . In terms of 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 =©{ft' "• p i s a prime] whenever ft' i s an i r elementary r a d i c a l c l a s s ( r e c a l l that ftp = FC p fl ft') . I t f o l l o w s that i f ft' < PC then ft' = 0 ( f t p : p i s a prime] . In the f o l l o w i n g theorem we s h a l l prove that each ft^ must equal e i t h e r F C p or 3^ pYl(S) or ^ (S) f o r some C.U.D. set of in t e g e r s S . And conversely, any " d i r e c t sum" of such elementary r a d i c a l classes i s again an elementary r a d i c a l c l a s s . In other words, every elementary r a d i c a l c l a s s which i s contained i n PC i s a " d i r e c t sum" of these simple r a d i c a l classes (PC , 5" Yl(S) , 3" (S)) and a i l " d i r e c t sums" of such cl a s s e s are elementary r a d i c a l c l a s s e s . - 79 -3 . 4 . 1 4 THEOREM: For each prime p , tt' = FC or 3" f\(S ) or P P P * p y 3" (S ) f o r some C.U.D. set of in t e g e r s S i f tt' i s an P p' P elementary r a d i c a l c l a s s . A lso tt' fl FC =@(Wp : p i s a prime} Conversely, . tt' = © ^ [ p ] : P l s a P r l m e ) l s an elementary r a d i c a l c l a s s i f f o r each prime p , ttj. ^ = FC p or 3 yi(S ) or 3" (S ) f o r some C.U.D. set of Integers p ' ^ p' P P S . Moreover, M = Mr , f o r a l l primes p . P P [p] Proof: Let tt' be an elementary r a d i c a l c l a s s . Define S = (n : F n e tt'} . Then S i s a C.U.D. P P P set of p o s i t i v e i n t e g e r s since tt' i s s t r o n g l y h e r e d i t a r y . We must show that ttp = FC p fl tt' i s FC p or ^ ( S p ) ° v 3 yi(s ) . p p ; I f tt' = FC we are done so suppose tt' / FC P P v P P We w i l l consider the two cases of 3 - 4 . 1 3 -I f »' n 7 l = ( ( 0 ) } we w i l l show that tt' = 3" (S ). p v y P p p Suppose that R e tt' = FC n tt' and that x i s a P P non-zero element of R . Then Y\(<x>) = ( 0 ) since tt^ fl ft = { ( 0 ) } . Let P be a prime i d e a l of <x> •. Then <x>/P must have c h a r a c t e r i s t i c p so e i t h e r <x>/P Is f i n i t e or <x>/P ~ F [X] where X i s an indeterminate. I f <x>/P =.Fp[X] then every r i n g of c h a r a c t e r i s t i c p i s i n M' (since they are a l l homomorphic images of F [X]) . - 80 -But then FC p <_ ft' because i f A Is a r i n g and B <5 A such that A/B e ft' and B e ft' then A e ft' . Hence ftp = FC p. This i s contrary to our s u p p o s i t i o n that ftp / FC p . Hence <x>/P i s f i n i t e . Then <x>/P i s a Wedderburn r i n g without zero d i v i s o r s so i f P / <x> then <x>/P i s a f i e l d and hence P i s a maximal i d e a l of . <x> . Since <x> s a t i s f i e s A.C.C. and a l l prime i d e a l s are maximal, <x> s a t i s f i e s D.C.C. (Theorem 2, page 203 of Z a r i s k i and Samuel [ 1 5 ] ) . Therefore <x> i s a commutative Wedderburn r i n g so <x> = F a, © ... © F av . Then P a . e ft' so the a. e S p l p K p i l p Therefore R e 3" (S ) . P P Conversely, I f R e ^p^p) t h e n f o r a 1 1 x e R , <x> i s a f i n i t e d i r e c t sum of f i e l d s i n ft' so <x> e ft' . P P Therefore R e ft' , so ft' = 3 (S ) . P ' P P P I f J! n PC < ft' we w i l l show that tt' = 3 *W(S ). uv, p _ p p p " 5 V p ; Suppose R e ftp = PCp n ft' and x e R . Then as above <x>/Jl(<x>) e ffp(Sp) . Since R e ftp , R e FC p so R e ? p H ( S p ) . Conversely, i f R e 3 p T l ( S p ) then f o r a l l x e R , <x>/*|i^ (<x>) i s a f i n i t e d i r e c t sum of r i n g s i n ft' so 3? <x>/y|(<x>) e ft^ . Since R e 3 p Y l(S p) , R e PC p so Yl(<x>) e PC p f\fl < ftp . Therefore <x> e ft^ so R e ftp . Hence ftp = %Yl(Sp) . - 81 -Prom 3 ' 4 . 9 we know that 34' n FC = © [ M p ': p i s a prime} . . We s h a l l now prove the converse. Assume that 34 = Q {Jij-p-j : p i s a prime} ivhere 34j-pj = FC p or 3p"y2(S or 3" (S ) f o r some C.U.D. set of i n t e g e r s S f o r each p p P prime p . Since w [ p ] ^  ^ f ° r a-L-5- P r i m e s p , ti < FC . F i r s t we s h a l l prove that ti = ti' . Suppose R e and l e t x e R . Since R e ti , R = © [34j- pj(R) : p i s a prime} . Therefore x = x 1 + ... + x n where x i e 341- -j (R a . 1 and P i 1 x ± = 0 f o r some Integers a± > 1 . Moreover, i f i ¥ 3 , Pj_ ¥ P j • Thus, f o r each i , there i s an i n t e g e r d. such that d.x = x. . Since ( x . ) . v e'FC , i i i v i ;<x> p i 5 ( x i ) < x > 0 . S . ( x i ) < x > = ( 0 ) * T h e r e f o r e , jV i <x> = ( x 1 ) < x > + ... + ( x n ) < x > . Since H [ p j i s s t r o n g l y h e r e d i t a r y , ( x i ) < x > e **[p.] ' Therefore, <x> e 34 ; so R e 34' . Suppose R e 34' . Then R e PC so R = © {FC p(R) : p i s a prime} . We s h a l l show that FC p(R) = M[ p](R) • C l e a r l y M [ p ] ( R ) £PC p(R) . Let x e FC p(R) . Since R e 34' , <x> e 34 . Therefore, <x> = © {34j-p-j (<x>) : p i s a prime} . But <x> e FC p so - 82 -M [ q ] ( < x > ) = (0.) f o r a l l q / p . Thus, <x> = M [ p ] ( < x > ) so <x> e tt[p] . Since tt[p] = tt'^ , FC p(R) e tt[p] . Therefore, FC p(R) = W [ p ] ( R ) so R e t t . Hence tt = tt' i s an e l e -mentary c l a s s . Notice that we have shown that tt = FC fl tt c ttr , so c l e a r l y =Tttr , . P P ~ LPJ J P Lp] Suppose R e t t and l e t R' = R/I be a homomorphic image of R . Then M [ p ] ( R ' ) - ^ [ p ] ^ ) + s 0 c l e a r l y R' = © ^ [ p ] ( R / ) : p i s a' prime} . Therefore 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 A i s a r i n g and B < A such that A/B and B e tt . Since both A/B and B are i n FC , A e FC . Therefore, by 3 . 1 . 6 ( i i i ) A = © (FC p(A) : p i s a prime} . (*) Since B e tt , FC p(B) = W [ p ] ( B ) and since both FC p and tt|-p-j are h e r e d i t a r y (see Theorem 4 8 i n Di v i n s k y [ 7 ] ) , tt[p](A) n B = « [ p ] ( B ) = FC p(B) = FC p(A) fl B . (**) . Now (FC p(A)/FC p(A) n B) = (FC p(A) + B/B) c P C p ( A / B ) e tt[p] since A/B e tt . By (**), FC p(A) D B e tt^j , so since tt^ i s a r a d i c a l c l a s s , FC p(A) e ttj-p-j . C l e a r l y * * [ p ] ( A ) E F C p ( A ) - 83 -so M [ p ] ( A ) = F C p ( A ) • N o w (*) i m p l i e s that A e ft . Therefore, by 2 . 3 . 1 , ft i s an elementary r a d i c a l c l a s s . Q.E.D. The f o l l o w i n g l i s t provides a r e p r e s e n t a t i o n of each of the elementary r a d i c a l classes ft <_ FC which we have al r e a d y discussed. Let Z be the set of a l l p o s i t i v e i n t e g e r s . = © (3p(.Z +) : p i s a prime] . = £>' = W n FC = Q {X 71(0) : P i s a prime] . 0 P C p = ^ ( z + ) • 71 n pc p = ^71(0) • <% n FC =Q {3 p 7 2(Z +) : p i s a prime] . The r e l a t i o n s h i p s between the elementary r a d i c a l c l a s s e s which we have discussed can be i l l u s t r a t e d by the f o l l o w i n g diagrams. - 84 -ILLUSTRATION 1 - 85 -CHAPTER IV GENERALIZED ELEMENTARY AND LOCAL RADICAL CLASSES 4.1 ABSORBENT CARDINAL NUMBERS. In t h i s f i r s t s e c t i o n of Chapter IV we s h a l l prove that 2 and |^ are absorbent c a r d i n a l s . In f a c t , we s h a l l show that any c a r d i n a l K such that 2 <_ K <_ f-f i s absorbent. We begin w i t h the f o l l o w i n g two lemmas. 4.1.1 LEMMA: I f R 4 (0) and R i s a zero r i n g (R 2 = (0)) of c h a r a c t e r i s t i c p f o r some prime p then R can be homo-mor p h i c a l l y mapped onto Cp = the zero r i n g on the c y c l i c group of p elements. Proof: Let R be a r i n g such that R / (0) , R 2 = (0) and pR = (0) . Choose x e R , x / 0 . By Zorn's Lemma choose K maximal i n Z = ( 1 4 R : x | 1} . Suppose 0 -/- w e R and w k K . Then, x e (w) R + K so there i s a non-zero i n t e g e r n such that x = nw + y when y e K . Since pR = (0) c K and x ^ K , p does not d i v i d e n . Therefore, there are in t e g e r s r and s such t h a t rp + sn =. 1 ; so, w = (rp + sn)w = snw = sx - sy . Thus, w e <x> c R/K so R/K = <x> =• Cp . Q.E.D. - 86 -Let p be a prime number. The r i n g p i s d i s -cussed i n Rings and R a d i c a l s . D i v i n s k y [7] • We may th i n k of t h i s r i n g as the set of a l l r a t i o n a l numbers of the form — where p does not d i v i d e a . The a d d i t i o n i s modulo 1 n ^ P and the 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 isomorphic to Cp = the zero r i n g on the c y c l i c group of p elements f o r some n , and a l l non-zero homomorphic images of p r a are isomorphic to p M . 4.1.2'LEMMA: I f R i s a zero r i n g and there i s an x e R such that x / 0 but px = 0 f o r some prime p then R can be homomorphically mapped onto Cp or onto p r a . Proof: Let R be a r i n g such that R = (0) and l e t x e R such that x / 0 but px = 0 .• By Zorn's Lemma choose I maximal i n Z = ( I < s Q R : x < ^ I } . I f p(R/I) / R/I then R/I can be homomorphically mapped onto a zero r i n g of c h a r a c t e r i s t i c p so by Lemma 4.1.1 R can be homomorphically mapped onto Cp . Suppose p(R/I) = R/I . Let w e R and w £ I . Then x e (w) R + 1 so there i s a non-zero i n t e g e r n such that x - nw e I . Therefore pnw e l so R/I e FC . Now FC (R/I) i s a d i r e c t summand of R/I and hence a homomorphic - 87 -image of R/I . Therefore, PC (R/I) = R A f o r some i d e a l K of R and since x e PC (R/I) , x £ K .. By the maximali-t y of I , FC (R/I) = R/I . Let w e R/I and n be a p o s i t i v e i n t e g e r . Write n = p n where p does not d i v i d e n . Since p(R/I) = R/I , p a ( R / I ) = R/I so there Is an element v e R/I such that p°v = w . Now, R/I e FC so there i s an i n t e g e r • ie— — z k such that p w = 0 . Since p does not d i v i d e n there are Integers r and s such that r p k + sn' = 1 . Then w = (rp + sn )w = sn w = sn p v = n p (sv) = n(sv) . There-f o r e , the a d d i t i v e group R / I + i s d i v i s i b l e . Since (R/I) = (0) the i d e a l s of R/I are j u s t the subgroups of the a b e l i a n group R / I + . Therefore, 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 ] ) , R/I Is isomorphic to a d i r e c t sum of copies of p°° . Therefore R/I can be homomorphically mapped onto the r i n g . Q.E.D. We are now ready to prove the theorem. 4.1 . 3 THEOREM: I f K i s a c a r d i n a l number and 2 <_ K <_ |^ then K Is absorbent. Proof: Let K be a c a r d i n a l , 2 <_ K <_ j$ . Suppose that - 88 -I + IR + RI + RIR Suppose that M £ J g(K) Then there i s a non-zero homomorphic image M' = M/K' of M such that no K-subring of F i r s t we s h a l l prove th a t I / I | FC - Suppose I / I e FC . N Let x e I . then mx = S u.v. where u. , v. e I and m i r = i 1 1 1 1 i s a non-zero Integer. Now, there Is a non-zero i n t e g e r k 2 such that ku. e I f o r 1 = 1 , .... n . Therefore (km)x e I 3 , so I / I 3 e FC . But then i f w e M = ( l ) R , nw e M 3 c K ' f o r some i n t e g e r n / 0 . This i s impossible because z e M/K' and <z> \ FC . Therefore I / I £ FC so 2 I / I can be homomorphically mapped onto a non-zero r i n g L such that L = (0) and L has c h a r a c t e r i s t i c 0 ( f a c t o r out F C ( I / I 2 ) ) . Choose x e L , x 4 0 and by Zorn's Lemma choose H maximal i n the c l a s s Z = { J < J L : i f n i s a non-zero i n t e g e r then nx J } . Then FC(L/H) = (0) , f o r I f y e L, y \. H but ny e H f o r some non-zero i n t e g e r n then by the maximality of H , kx e ( y ) ^ + H f o r some non-zero i n t e g e r k so (nk)x e H which c o n t r a d i c t s the way i n which H was - 89 -chosen. Let S be a K-subring of L/H . Since K <_ ^ , S Is f i n i t e l y generated. Because L/H i s a zero r i n g the i d e a l s of S are j u s t the subgroups of the a d d i t i v e group S' Since S i s f i n i t e l y generated we may apply the fundamental theorem f o r f i n i t e l y generated a b e l i a n groups to see that S i s isomorphic to a f i n i t e d i r e c t sum of copies of C°° . I f u and v are two non-zero elements of L/H there are non-zero i n t e g e r s k and n such that kx e ( U ) L + H a n d nx e ( v ) ^ + H . Hence there are non-zero i n t e g e r s r and s such that kx - r u e H and nx - sv e H. This insures that the d i r e c t sum i n the preceding paragraph i s of le n g t h 1 ; that i s , S * C°° . Now, since I e $ g(K) ' s o m e K-subring of L/H (which i s a homomorphic image of I) i s i n el . Since a l l K-subrings of L/H are Isomorphic to C05 , 0°° e o$ . This i s a c o n t r a d i c t i o n s i n c e we assumed that M/K7 d i d not contain a K-subring. Case 2: There i s a z e M such that Cp = <z> c MA' f o r some prime p . 2 2 2 I f p ( l / I ) / I / I then I / I can be homomorphi-c a l l y mapped onto a zero r i n g L / (0) of c h a r a c t e r i s t i c p. 2 By Lemma 4.1.1, I / I can be homomorphically mapped onto Cp . So, s i n c e I e «5 , Cp e J . This i s a c o n t r a d i c t i o n S ( K ) - 90 -because we have assumed that M/K7 has no K-subrlngs i n 2 2 We must conclude then, that p ( l / I ) = I / I Since z e M = ( l ) R there are i n t e g e r s nu and elements x. e I and v., s., v., s. e R such that l . i ' i ' i ' l L _ _ z = S r.x.s. + r.x. + x.s. + m.x. . (*) i l l i i i i i i v / Now, p ( l / I 2 ) = I / I 2 , so p i + I 2 = I . Hence I 2 = I ( p l + I 2 ) = p i 2 + I 3 so I = p i + I 2 = PI + p i 2 + I 3 p i + I 3 . Thus p n ( l / I 3 ) = I / I 3 f o r a l l p o s i t i v e i n t e g e r s n . For each n >_• 1 choose x. e l such that n T 3 l n ' x. - p x. e l . i ^ I n L Let z = E r.x. s. + r.x. + x. s. + m.x. n . , 1 1 1 i i i i i i . 1=1 n n n n Then z - p n z e M 3 <= K 7 so z" = p nz" / 0 i n M/K7 . Moreover, since pz = 0 , p " z n = 0 so < z n> = Cp i n M/K7 Now, suppose that f o r a l l x e I , x e ( p x ) T + I 'I Then i f x e I , there i s an i n t e g e r n such that 2 2 2 x - pnx e l so ( l - pn)x e I . Therefore I / I e FC . Suppose F C p ( I / I 2 ) = (0) . Let x e I . Then there i s a non-zero i n t e g e r r such that p does not d i v i d e H 2 r and r x = Y. u.v. e I . Now there i s a non-zero i n t e g e r 1=1 1 1 p s which i s not d i v i s i b l e by p such that su^ e I f o r 1 = 1 , H .• Thus, srx e I and p does not d i v i d e - 91 -s r . Therefore, we may choose a non-zero i n t e g e r k which i s not d i v i s i b l e by p and such that kx^ e I • f o r i = 1, ..., L . Now from (*), kz e ( l 3 ) R £ M 3 c K' . This c o n t r a d i c t s our assumption that <z> = Cp . Therefore F C p ( l / I 2 ) / (0) so by Lemma 4.1.2 2 05 I / I can be homomorphically mapped onto Cp or onto p tha On the other hand, i f there i s an x e I such 2 2 t x £ ( P X ) I + I then I / I can be homomorphically 2 mapped onto I / ( ( p x ) j + I ) which by Lemma 4.1.2 can be homomorphically mapped onto Cp or onto p"5 . Thus our assumption i n Case 2 leads to the conclu-s i o n that I can be homomorphically mapped to Cp or to •gT . We have seen that the conclusion that I can be homo-mo r p h i c a l l y mapped to Cp leads to a contraduction. Now, i f I can be homomorphically mapped onto p r a, CO P e 520 ~ • Since a l l K-subrings are f i n i t e l y generated, S(K) Cp n e f o r some p o s i t i v e i n t e g e r n . But then <z n_ 1> e i f n _> 2 and <z> e g(f i f n = 1 . In any case t h i s c o n t r a d i c t s our assumption that M / K ' contains no K-subring i n E i t h e r Case 1 or Case 2 must occur since ( M / K ' p = (0) . Both cases lead to a c o n t r a d i c t i o n so we conclude that M e J( . Therefore K i s an absorbent Q.E.D. g(K) c a r d i n a l . - 92 -^•2 GENERALIZED RADICAL CLASSES WHICH ARE 4 fj . O We s h a l l r e f e r to ge n e r a l i z e d 2-classes as g e n e r a l i -zed elementary 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 to as ge n e r a l i z e d l o c a l c l a s s e s . We s h a l l w r i t e ti o f o r ti i it \ and ti f o r ti ( , . The f o l l o w i n g p r o p o s i -t i o n shows that t h i s w i l l not 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 of Andrunakievic and T h i e r r i n . 4 . 2 . 1 PROPOSITION: o Proof: Assume that R e ~fl. • Let 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' . Therefore R eY? g (2) I f R e * ^ g ( 2 ) then every non-zero homomorphic image of R contains a n i l subring so c l e a r l y no non-zero homomorphic image of R i s fl s.s. Therefore, R e 77 g g By 2 . 4 . 2 ( v i i ) , 7^(2) = ) so Q.E.D. 4 . 2 . 2 THEOREM: I f ti i s any c l a s s of r i n g s such that j3 <_ ti <_ FP - 93 -then tt = Y\ g 1 § Proof: Assume that tt i s a c l a s s of r i n g s and £ <_ tt- <_ FF, Since *y| =71 , by 2 . 4 . 2 ( i i ) we need only show that g g-j^  > Tig and FF < 7 ? g ' Since a r i n g generated by one element i s i n j3 i f and only i f i t i s i n 71 I t i s c l e a r that 6 = Yl = 71- • s l s l . s Let R be a r i n g such that R £ Y( . Then there g i s a non-zero homomorphic image R' of R such that R' has no non-zero n i l p o t e n t elements. Thus, f o r a l l 0 / x e R' , <x> i s 71 semi-simple. In Theorem 3«3-2 we proved that such a r i n g <x> could be homomorphically mapped onto a f i n i t e f i e l d . Hence, no non-zero subring <x> of R 7 i s i n FF , so R I PP . Therefore, FP < g-L g 2 ~ Q.E.D. I f R e "Yl then every non-zero homomorphic image of R contains a subring <x> such that <x> i s n i l p o t e n t . Thus, R e jS <_ jS (by 2 . 4 . 2 ( i i i ) ) . Now, by 2 . 4 . 2 ( 1 1 ) we g-^  g must have yi = «£ = j3 'LS g g Using 2 . . 4 . 2 ( i i ) again we see that 7) < J < J g $ < 3 ' < F < F F . Almost a l l questions con-y i g ~ g - g - g - g - g cerning these r a d i c a l c l a s s es are open. We do not even know which of the above i n c l u s i o n s are s t r i c t . Notice however, - 9k -that FF < PP since subrings of f i n i t e f i e l d s are f i n i t e g - b f i e l d s . 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 -zed classes a s s o c i a t e d w i t h oSr and % ^ • Notice that by 2 . 4 . 2 ( v i i ) & g = * and' ( & R ) g = ( & R) • 4.2.3 PROPOSITION: ( i ) R e & g i f a n d o n l y i f R / ( & R ) g ( R ) e PC • ( i i ) * . g * C f r R ) g * T i g • ( i i i ) R i s %- semi-simple i f and only i f f o r a l l x e R, <x> = fax] • Proof: ( i ) Assume that R e & . I f R / ( & R ) _ ( R ) $ FC l e t I O R such that I = ( & v ) (R) and I / ( # R ) ( R ) = FC(R/( & R) ~(R)) • Since R/I e £ there i s an f g g x e R such that x £ I and a n x n + ••• + a^x e I f o r some i n t e g e r s a^, a n . Then there i s a non-zero i n t e g e r m such that ma x n + ... + ma nx e (&„) ( R ) .. Let b. = ma. (ma ) n _ 3 n 1 v R/gx ' I 1. rv' and l e t y = ma x . Then J n n . . . n-1 y + b n - l y + ... + b i y € ( l R ) g ( R ) • ( f ) Since x £ I , y ^ I so c e r t a i n l y y <t ( & R) (R) . We - 95 -_ <y> + ( & R ) , ( R ) ^ w i l l prove that <y> = — ( vg. ) ( R ) i s l n ( J r R ) g -By (j>), the a d d i t i v e group <y> + i s f i n i t e l y generated so I f w € <y> , <w> i s a l s o a f i n i t e l y generated a b e l i a n group. Thus there are polynomials f 1 ( w ) , f K ( w ) which generated <w>+ . Choose an i n t e g e r h which i s l a r g e r than the degree of each h L f. . Then w = S b.f.(w) f o r some i n t e g e r s b^, - b g . Therefore, <y> e R Since Jjr-^ s a t i s f i e s c o n d i t i o n (A), <y> e ( X*R)g • Now since f>cQ i s absorbent (see 4 . 1 . 3 ) the non-zero i d e a l of R / ( $ R ) (R) which i s generated by <y> i s i n ( ^ R ) G • This i s a c o n t r a d i c t i o n . Hence R/( & R ) (R) e PC . Both PC and ( „ are <_ %• so the g g converse i s obvious. ( I I ) C l e a r l y % g > ( & R ) g > ) t g • The r i n g P p [ X ] e ^ but i s not i n ( & R ) g so & ¥ (** R) g- N o f i n i t e f i e l d i s i n y\ but they are a l l i n ( fr-n) SO (&R)g ¥ yig • ( i i i ) This f o l l o w s immediately from 2 . 4 . 1 5 and 3 . 1 . 3 -Q.E.D. We s h a l l now consider the classes ((&- R) )' and - 96 -( )' . By 2.4.2(ix) both classes are r a d i c a l c l a s s es and c l e a r l y ( ( & R ) g ) ' = (( £ R ) g ) * and ( & g ) ' = ( £ g ) * . Since $y and & R are elementary r a d i c a l c l a s s e s / ( \Y > & and ( ( & R ) g ) ' > % R • I f R e ( ^- ) ' and x e R then <x> e ^ so g o <x> % (p[X] , thus <x> 6 jjr so R e tr • Therefore, ( &g) = • Now £ R < ( * R ) g < & g so & R < ( ( & R ) g ) ' < 4.2.4 PROPOSITION: ( 1 ) P C n ( ( & R ) g ) ' < ^ R (2) R e ( ( & R ) g ) ' i f and only I f ' f o r a l l x e R , <x>/J»R(<x>) has c h a r a c t e r i s t i c 0. and i s i n *tr Proof: ( 1 ) Suppose R e PC fl ( ( & R ) ) ' . Then R = £) ( P C (R) : p i s a prime] so sin c e e f r R i s a r a d i c a l c l a s s i t i s s u f f i c i e n t to show that PC p fl ( ( « f r R ) g ) ' <L £- R' f o r each prime p . Let A e PC p fl ( ( & R ) g ) ' and l e t x be a non-zero element of A . Then there are i n t e g e r s a^, and a y = a^x + ... + a^x p<x> such K K - 1 that y + b K_^y + ... + b^y e p<x> f o r some in t e g e r s b K - l ' '''' b l ' W e m a y a s s u m e that p does not d i v i d e TT } XT 9 _1 a^ . Thus x^ = x + ^ Y i l - 1 x + ... + c n x e p<x> - 97 -f o r some in t e g e r s c^, . .., c K - t - i * B y r e p e a t i n g the above argument we see that f o r each i n t e g e r n _> 1 there i s a monic polynomial i n x which i s ' i n pn<x> Ci Since p x = 0 f o r some i n t e g e r a , we- conclude that Assume that R e ( ( J r R ) ) ' and l e t x e R . Then s i nee < x > / c t ? R ( < x > ) i s $ y R semi-simple, by part ( l ) <x>/<£R(<x>) must be of c h a r a c t e r i s t i c 0 . Since ((^R)g)/ < t > <*> € tr • Conversely, assume that f o r a l l x e R , <x>/e1rR(<x>) e i r and has c h a r a c t e r i s t i c 0 . Let x e R such that <x> = <x>/^ R(<x>) / (0) . Then <x> e I- so a^jc^" + . . . + a-^ x = 0 f o r i n t e g e r s a^, a^ Since <x> has c h a r a c t e r i s t i c 0 we may assume that K >_ 2 and that the greatest common d i v i s o r of the a^ i s 1 . Now i f y = a^x , y K + a K - l y K - 1 + ^ a K a K - 2 ^ y K ~ 2 +••••+ ( a K K _ 2 a l ^ y = 0 ' This guarantees that <y> e ( $ R ) . Therefore, since O i s absorbent, ( y ) < ~ > e ( £ R ) g • Now, since a K x = y e ^y^<x> a n d t n e S r e a t e s t common d i v i s o r of the a i i s 1 , the r i n g < x > / ( y ) < x > i s f i n i t e and hence i n ( & R ) g • Thus, <x> e ( £ - R ) g • Therefore, <x> e (J&-R) so R e ((trR)s)' . Q.E.D. - 98 -I t f o l l o w s Immediately that Jj- R By 2 . 4 . 2 ( v i i ) ( ( C t R ) g ) ' ) g = ( £ R ) g • 4 . 3 GENERALIZED RADICAL CLASSES WHICH ARE 4 FC . 4 . 3.1 LEMMA: Let p be a prime. Then ( F C p ) g ( K ) = F C p f o r ' a 1 1 c a r d i n a l numbers K >_ 2 . Proof: Let p be a prime and K be a c a r d i n a l >_ 2 . Since PC 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 s t r o n g l y h e r e d i t a r y , PC p <_ ( p c p ) g ( K ) * Suppose R e ( F C p ) g ( K ) . Let R = R/PC p(R) . I f R -/ (0) then since R e (FC ) there i s a non-zero P g(K) K-subring S of R such that S e F C p . But then ( S ) R e PC p which i s a c o n t r a d i c t i o n since R i s PC semi-simple. _ P Hence R = (0) so R e FC v ' P Q.E.D. 4 . 3 . 2 PROPOSITION: Let tt be an elementary r a d i c a l c l a s s <_ FC and l e t M = © ( W : p e S} be the r e p r e s e n t a t i o n of tt given XT i n Theorem 3• 4.14. Then f o r any c a r d i n a l K >. 2 , R 6 ^o-(K) - 9 9 -i f and only i f R = © ( ( M p ) g ( K ) ( R ) : P e S) . Thus we may denote ttg^ by © C ( M p ) g ( K ) : P e S] . Proof: Let S be a set of primes, tt (tt p : p e S] , and l e t K be a c a r d i n a l > 2 . Since ttp = tt n PC p <_ FC p , (W ) /Tr\ < (PC ) t v \ = p c • Thus, f o r any r i n g R , the sum v p'g(K) - v p'g(K) p of i d e a l s ( M p ) g ( K ) ( R ) i s d i r e c t . Suppose R e M g ( K ) • Then R e P C g ( K ) a n d J' u s t a s i n the proof of 4 . 3-1, R e FC . Hence by 3-1-6 R = © ( P C p ( R ) : p i s a prime] . Now, f o r any prime p , FCp(R) i s a homomorphic image of R so since R e ^ ( K ) > PC p(R) = (0) i f p <t S . Thus, R = © (FC (R) : p € S] . Let p e S and l e t R be a non-zero homomorphic image of FC (R) . Then R i s a homomorphic image of R e t t ,Trs so there i s a non-zero K-subring H c R such that g(K) H e tt . Now H e tt fl FC = W so PC (R) e (U) , . . By p P P P g(K) 4 ' 3 ' 1 ( V g ( K ) ^ F C p 3 0 • F C p ( R ) . = ( V g ( K ) ( R ) ' T h e r e f o r e > R =Q C(»p)g(K)(R) : P e S) • Conversely, suppose th a t R = © K ^ p ) g ^ ^ (R) : P e S) Since tt < tt f o r a l l p e S i t i s c l e a r that R e t t f v* . P.— g ( K ) This completes the proof. Q.E.D. - 100 -Combining 4 . 3 - 1 and 4 -3.2 we see that i f .S i s any-set of primes then tt = © ('FC : p e S] i s a ge n e r a l i z e d K-class f o r a l l c a r d i n a l s K > 2 . In f a c t . M ( v s = tt . — 3 g ( K ) Now tt i s a l s o a K - c l a s s . In the next theorem we s h a l l show that these are the only classes which are both K-classes and ge n e r a l i z e d K-classes (except; of course, f o r the c l a s s of a l l r i n g s ) . 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 cla s s of r i n g s which does not contain a l l r i n g s . Then ft i s a s t r o n g l y h e r e d i t a r y g e n e r a l i z e d K-class i f and only i f ft = © ( F C p : p e S] f o r some set of prime numbers S . Proof: Assume that S i s a set of prime numbers and ft = Q (FCp : p e S) . By 3 - 4 . 1 4 ft i s an elementary r a d i c a l 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 . From 4 . 3 . 1 and 4 .3.2 ft Is a gen e r a l i z e d K - c l a s s . Conversely, assume that ft i s a s t r o n g l y h e r e d i t a r y g e n e r a l i z e d K - c l a s s . Then there i s a c l a s s of r i n g s 3" such that ft = ^ ( K ) * Since 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 ) , (* g ( K ) ) (2) < ( * g ( K ) ) g ( 2 ) < * g ( K ) • *hus » < -B(2) = (3g ( K ) )(2) < ^ g ( K ) ) g ( 2 ) ^ *g(K) = R ' H e n C e ft = ft(2) = ( 3 ' g ( K ) ) g ( 2 ) = ^g(2) ' N o w w e s h a 1 1 Prove that i f - 101 -ft FC then ft is the c l a s s of a l l r i n g s . Suppose ft <t_ PC, then there i s a r i n g R e f t such that PC(R) = (0) . F i r s t we s h a l l prove that c " e ft . Let 0 4 x € R . Then <x> e f t so <x>/<x>2 e ft . 2 oo 2 I f <x>/<x> % C then nx e <x> f o r some i n t e g e r n . Thus, there are in t e g e r s a 2 , ..., a^ such that 2 K K —1 nx = a^x + . ..- + a^x . Let y = a^x + .. . + a^x Then nx = yx so <y> = n*Z = the i d e a l of the in t e g e r s generated by n . Since ft = ft(2) , <y> e ft • Now consider . Y = (<y>) 2 = the r i n g of 2x2 matrices w i t h e n t r i e s from <y> . Then every non-zero homomorphic Image of Y contains a subring generated by one element which i s Isomorphic to a homomorphic image of <y> . Therefore Y e R g ( 2 ) = B ' Hence, C™ = <(° g ) > e ft(2) = ft . So i n any case C03 e ft . Since c " e ft and j3 = the lower r a d i c a l c l a s s determined by {0°°} , j8 <. ft . Thus 71= 0(2) <_ ft(2) = ft . Let Q(X) be the f i e l d of 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 X over Q = the f i e l d of r a t i o n a l numbers. Then ( Q ( X ) ) 2 i s a simple r i n g w i t h non-zero n i l p o t e n t elements. Thus ( Q ( X ) ) 2 e & g ( 2 ) = R ' B u t t h e n , since ft = ft(2) , ( p [ X ] = < ( Q Q)> e ft • Then a l l r i n g s generated by one element are i n ft since ft s a t i s f i e s " c o n d i t i o n (A). There-f o r e ft = ft / 0v = the c l a s s of a l l r i n g s . g(2) Since ft i s not the c l a s s of a l l r i n g s , ft <_ PC . - 102 -Let S = [p : ftp / {(0)}} . Then, by Theorem 3-4.14, ft = ® (ftp : p e S} . We w i l l show that Fp[X] = the r i n g of polynomials i n an Indeterminate X over Fp i s i n ftp and hence ft = FC f o r a l l p e S . p p . Let (0) 4 <x> e ft . Then i f <x>/<x>2 has a ir non-zero n i l p o t e n t element, Cp = the zero r i n g on the group of p-elements i s i n ftp . I f <x>/<x> has no non-zero n i l p o t e n t elements then <x> has an i d e n t i t y so Zp e ftp . In t h i s case ( Z p ) 2 e 'ftp <_ ft so sin c e ft(2) = ft , Cp e ftp . So i n any case' Cp e ftp . Now ( F p [ X ] ) 2 e = & s o Fp[X] e ft(2) = & • Therefore every r i n g of 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 of a r i n g A and both A/B and B are i n ft then A e ft ._ Hence FC < ft . P ' P P - P Therefore, ftp = FC p f o r a l l primes p e S . This completes the proof. Q.E.D. We now turn to a c o n s i d e r a t i o n of some cla s s e s of r i n g s W such that it =7? n FC . g " lg 4 . 3 - 4 PROPOSITION: I f R e £> and R e J . R i s f i n i t e l y generated then - 103 -Proof: Let R be a f i n i t e l y generated r i n g i n S) . I f R e PC then R e J3 fl PC so R e J by 3-2.1. Suppose that R £ PC . Let R = R/PC(R) . Since R e ^ <_ <£ , there i s a p o s i t i v e i n t e g e r n such that R n 4 (0) but R^+1 = (0) . Since R e f ) , there i s a r i n g A w i t h D.C.C. on l e f t i d e a l s such that R = Yl(A) . Now R n i s f i n i t e l y generated as a subring of R by 2.2 . 6 . Moreover, W1 has c h a r a c t e r i s t i c 0 so from the Fundamental Theorem f o r f i n i t e l y generated a b e l i a n groups R i s isomor-phic to a f i n i t e d i r e c t sum of copies .of C™ . But then 2 R n 2 2 R n p ... P 2 kR n ? ... and sin c e 2 kR n<3 A f o r a l l p o s i t i v e i n t e g e r s k t h i s c o n t r a d i c t s the D.C.C. c o n d i t i o n f o r A . Thus we must have R e FC so R e J . Q.E.D. Since J1 < D i t f o l l o w s immediately from 4 . 3 . 4 t h a t sS l - *8 l a n d »e - 4 • Let R be a f i n i t e l y generated r i n g i n <© and l e t x e R . Now, R e J = j6 n PC so R i s n i l p o t e n t and i n PC. Therefore <x> e jS fl PC = S so <x> eh- I t f o l l o w s that Notice that P r o p o s i t i o n 4 . 3 - 4 a l s o i m p l i e s that = j>* and fi' = j>' . - 104 -3" Is equal to any of the or • Proof: We have already n o t i c e d that % .= f = . A = J Since J < F C , J g < FC and since / < jB , J g < j3 g = 72 g • Thus J g < ^ g n PC- . Let R e || fl FC and suppose that R i s a non-zero homomorphic image of R . Then there i s an 0 / x e R such that <x> e FC and x i s n i l p o t e n t . Thus <x> e f (and <x> e J - * ) . Thus ^ g n FC < J& so / g = f\g fl FC. By 2 . 1 . 3 ( i i i ) J* < J' • Then by 2 . 4 . 2 ( i i ) , ( j f * ) g <-(J ' / )g i and ( / * ) g < ( J ' ) g and by 2 . 4 . 2 ( i i i ) , ( ; f * ) g < ( i * ) g a ^ ( S')&1 < ( C f ' ) g • Notice that i n the above paragraph we a c t u a l l y proved that Ylg H FC <_ ( Now combining these i n c l u s i o n s we have Tig n FC < ( / * ) g i < ( j > * ) g < ( j f ' ) g and 1 npc< ( f * ) < ( ! % < ( / ' ) „ • Since 8 * = / * and B ' = J " we need only show that ( f') < Jl PI FC o o to complete the proof. Since j 5 <. FC , ( J') < (FC') = FC and since g g S < Yi > J' < 71 s o (f)cr < Tip. • Therefore 4 . 3 - 5 THEOREM: .3" = 3" = 7 | HFC i f g g 1 " g c l a s s e s fc, , £ * , J * , £ ) ' - 1 0 5 -Q.E.D. In view of.-3.4.14 and. 4.3-2 to determine tt f o r o any elementary r a d i c a l tt which i s <_ PC i t i s s u f f i c i e n t to consider 3 where 3 i s one of the ba s i c b u i l d i n g g blocks of 3-4.14. 4.3-6 PROPOSITION: (1) ( 3 p ( S ) ) g = ( ( P p } ) g i f S 4 0 -(2) ( 3 - p T l ( S ) ) g = FC p n 72 g i f S = $ . (3) ( a r p Y l ( s ) ) g = P c p n ( & R ) g i f s / pr . Proof: Notice that since 3p(S) and ^ p 7 l ( S ) are ele-mentary c l a s s e s ( 3 (S.)) = (3" (S)) and ( J P W ( s ) ) S i - C p ™ 3 " , • ( l ) Suppose that S i s a non-zero C.U.D. set of p o s i t i v e i n t e g e r s . Since every non-zero f i n i t e l y generated r i n g i n 3" (S) contains a f i n i t e f i e l d of 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 that every non-zero homomorphic image of a r i n g which i s i n (S)) contains a non-zero idem-P g potent e such that pe = 0 . Thus (3 (S)) < ({P }) . \ p v J Jg _ u p J y g - 106 -Conversely, suppose that every non-zero homomorphic image of R contains an idempotent e 4 0 such that pe = 0 . Since S 4 P' and S i s a C.U.D. se t , 1 e S . Hence <e> = Fp e 3" (S) so R e (3 (S)) . p p g This completes the proof. (2) Since a: (0) = {(0)} , 3 71(0) - FC p Pi 71 and c l e a r l y (FC n 71 ) = FC f) Jl . (3) Suppose that S i s a non-zero C.U.D. set of p o s i t i v e i n t e g e r s . Now ^ p Y l(S) < Z ^ y i { Z + ) = FC p n i r R so c l e a r l y ( a T p V L ( S ) ) g <. PC p fl ( £ R ) g . I f R e FC fl (dSr-p) then every non-zero homomor-phic image contains a non-zero element x such that p a 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 1 a^, a^._^ I f x i s not n i l then <x>/7|(<x>) i s a commutative Wedderburn r i n g . Let y e <x> such that y = y + (<x>) i s the i d e n t i t y of <x>/Yl(<x>) . Then <y>/Yl(<y>) = Fp e J p ( S ) , thus <y> e 3" Yl(S) . So i n any case, R e (3 : p 7 2(s)) g . Therefore, (3Tp yi(S)) g = FC p n ( £ R ) g . Q.E.D. - 107 -k.h THE GENERALIZED RADICAL CLASS £ . g i A r i n g R e £ i f and only i f R 2 = R . Hence R e £ i f and only i f f o r each x e R there are elements N u. , v. e R such that x = E u.v. . Using t h i s c h a r a c t e r i -1 1 i = 1 i i z a t i o n of £ one e a s i l y sees that £ i s a r a d i c a l c l a s s . 2 Let R e £ . Since R/R cannot contain a non-g 2 . zero idempotent sub r i n g , R/R = (0) . Thus R e £ . By 2 . 4 . 2 ( i i i ) , £ < £ so £ < £ < £ . v g x ~ g g]_ g -Let R be the subring of the r i n g of r e a l numbers generated by the set of p o s i t i v e .real numbers { + ( 2 ) 1 / / 2 : n > 1} . C l e a r l y R 2 = R since [ ( 2 ) 1 / / 2 ] 2 = . l / 2 n / 2 ' . However, i f R 7 i s a f i n i t e l y generated subring of R then R 7 / (R 7) . This f o l l o w s from a lemma on page 215 of Z a r i s k i and Samuel [15] which i m p l i e s that i f (0) / I = I 2 and I i s f i n i t e l y generated 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 i m p l i e s that £ = £ f o r commutative r i n g s but we do not g^ g know i f £ (R) = £ (R) f o r a l l r i n g s R . s l . s Notice that R e £ i f and only i f every non-zero s l homomorphic image of R contains- an idempotent e / 0 , and that R i s £ semi-simple i f and only i f R has no idem-s l potent e / 0 . - 108 -C l e a r l y e ' < e so (e )' = (e )' = e ' . Of s l s l s course, (£'.) = (£') < FC n £ . Since the r i n g of g e1 sx i n t e g e r s moduls 4 i s not i n (£') , (£') £ FC fl £ . I f R i s a commutative r i n g and (0) / I i s a f i n i t e l y generated i d e a l of R which i s i n £ <_ £ then s l p I = I so I has an i d e n t i t y . . Thus I i s a d i r e c t summand of R . Therefore, f o r a commutative r i n g R wi t h A.C.C. & (R) has an I d e n t i t y and Is a d i r e c t summand of R . I f g l we do not i n s i s t that R be commutative we ob t a i n the f o l l o w i n g weaker r e s u l t . 4.4.1 THEOREM: I f ( O W R e & and R has A.C.C. on one-sided 2 i d e a l s then there i s an e e R such that e = e / 0 , R = ReR and e i s an I d e n t i t y f o r R/Yl(R) 4 (0) . Proof: Let (0) / R e £ and suppose that R has A.C.C. g l on l e f t i d e a l s . Then (0) / R = R/7^(R) i s semi-prime and so by Goldie's Theorem (Theorem 29 i n D i v i n s k y [ 7 ] ) R has a l e f t quotient r i n g Q, which i s a f i n i t e d i r e c t sum of 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 centre of Q so a l l iderapotents of R are i n the centre of R . Since R e £ there i s a non-zero Idempotent i n S l R . • Choose Rf maximal i n the set [Re : e Is an idempo-tent of R) . Since f e centre of R , Rf <3 R . If Rf / R then there i s an element w e R such that w i s a non-zero idempotent i n R/Rf . One e a s i l y checks that f ' = f +w - fw i s an idempotent and since f f ' = f , Rf c Rf' , By the maximality of Rf , Rf = Rf' . This implies that w e Rf which i s a contradiction. Therefore, Rf = R so f i s an i d e n t i t y f o r R . Then by Lemma 1.12 i n Herstein [10] there i s an idempotent e e R such that e + Yl(R) = f . Since Rf = R , ReR + N = R . Therefore R/ReR i s n i l and hence does not contain a non-zero idempotent. Thus R = ReR because R e £ S l The proof goes through i n exactly the same way i f we assume that R has A.C.C. on r i g h t i d e a l s . Q.E.D. The following proposition (together with Theorem 2.4.13) implies that i f M i s an elementary semi-simple class and M contains a l l £ semi-simple rings then M = the s l class of 3_ semi-simple rings f o r some class 5" which s.i s a t i s f i e s condition (A). - 110 -4 . 4 . 2 PROPOSITION: If 34 i s an elementary class of rings which con-tains a l l £ semi-simple rings then 34 s a t i s f i e s condi-s 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 class of rings which con-tains a l l £ s.s. rings. s l I f 34 s a t i s f i e s condition s"(2) then by 2 . 4 . 1 2 34 s a t i s f i e s condition s ( 2 ) . Suppose 34 s a t i s f i e s condition s ( 2 ) . By Theorem 2 . 4 . 9 34 i s a semi-simple class. I f <x> 34 then JJ(<X>) = I 4 (0) • Since 34 contains a l l £ g s.s. rings, ^ < £ . Therefore I e £ so by virtue of the remarks at the beginning of this section I has an i d e n t i t y and so there i s an i d e a l J of <x> such that I © J ' = <x> . Therefore I i s generated by one element as a subring of <x> and I cannot be homomorphically mapped onto a non-zero r i n g i n 34 . We have proven the contrapositive of condition s ( 2 ) . This completes the proof. Q.E.D. In the following proposition we show that i f Ji i s - I l l -an elementary semi-simple c l a s s which Is not of the type w i t h which 4 .4.2 i s concerned then ti must be contained i n the c l a s s of *)2 n ^0- semi-simple r i n g s f o r some prime p. S P 4 . 4 . 3 PROPOSITION: I f 3 be a' c l a s s of r i n g s which s a t i s f i e s c o n d i -t i o n r ( 2 ) then 3 >_ "ft H FC f o r some prime p or >1 • ' S l P r < e s i s i Proof: Let 2T be a c l a s s of r i n g s which s a t i s f i e s condi-t i o n r ( 2 ) . Suppose that f o r a l l primes p , 3_ £ Tin- H PC . g P Then f o r each prime p there i s a r i n g Rp e Yl fi PC which i s 3" s.s.. Since 3 s a t i s f i e s c o n d i t i o n r ( 2 ) , by s l Theorem 2 . 4 . 7 , the c l a s s of 3" s.s. r i n g s i s an elementary s l c l a s s . Therefore, f o r each prime p there i s a r i n g <x > c Rp such that <x > i s 3" s.s. and <x > = Cp . p - ^ p g x P F Since s u b d i r e c t sums of semi-simple r i n g s are semi-simple, 00 C i s 3 s.s. Now every zero r i n g on a c y c l i c group i s s l 3 s.s. so since the c l a s s of 3' s.s. r i n g s i s elementary g l s l and s a t i s f i e s c o n d i t i o n (P) a l l n i l r i n g s are 3^ s.s. I f R £ then there i s a non-zero homomorphic s l - 1 1 2 -image R of R such that there are no non-zero idempotents i n R . Let 0 / x e R and (0) / I <I <x> . Since there are no non-zero idempotents i n <x> , 1 / 1 . Thus I can be homomorphically mapped onto the non-zero 3" s.s. r i n g s l 2 I / I . Therefore <x> i s 3" s.s. Since the c l a s s of s l 3 s.s. r i n g s i s elementary, R i s 3 s.s. so R i J g-j_ g-j_ s l Therefore 3 <_ S . This completes the proof. s l s l 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 r a d i c a l c l a s s es i^hich we have discussed can.be i l l u s t r a t e d In the f o l l o w i n g diagrams. fC The f o l l o w i n g diagram remains v a l i d when PC i s everywhere replaced by FC 1? '3, £9j f\ FC ILLUSTRATION 2 - 114- -CHAPTER V LOCAL RADICAL CLASSES 5.1 THE RADICAL CLASSES ?.&.* AND £ . The c l a s s .* 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 -I f R e £ then R e 3 . © . * f o r i f R' = <x^, ..., x N> and R/K" = (0) then R' = (Sa.d. : d. e D and the a. are Integers} where ^ 1 1 1 1 D = (x. ... x. : L <_ K - 1} . However, f i n i t e f i e l d s are x l 1 L i n 3 . ft.* but not i n £ . Therefore, 3 . 8 .* ^ £ . Since £ ^ 7\ 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 that 3 . & . * i 71 . Hence 3 . 9 . * i s unrela t e d to % . 5.1.1 PROPOSITION: z = 3 . S .* n Yl . Proof: Since £ <_ 3.B .* and £ <_ 71 i t i s c l e a r that £ <_ 3 . & .* D "R . Assume R e 3.Sb .* n VI • Let R' be a f i n i t e l y generated subring of R . Then since R 7 e 3 . & .* , R' s a t i s f i e s A.C.C. on one-sided Ideals ( i n f a c t , the a d d i t i v e • group R / + has A.C.C. on subgroups). 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 sin c e a l l n i l p o t e n t r i n g s are i n 0 , £ <_ J3* . Therefore, £ = £* = |3* . 5 . 1 . 2 THEOREM: I f 34* i s a l o c a l r a d i c a l c l a s s and /3 <_ 34* <_ then a r i n g R i s 34* semi-simple i f and only i f R i s isomorphic to a su b d i r e c t sum of prime 34* semi-simple r i n g s . Proof; Let 34* be a l o c a l r a d i c a l c l a s s such that 0 ^ 34* <_ 3. & . * . Since s u b d i r e c t sums of semi-simple r i n g s are semi-simple one d i r e c t i o n i s c l e a r . Conversely, suppose that R i s 34* semi-simple. I t i s s u f f i c i e n t to f i n d , f o r each non-zero x e R an i d e a l l ( x ) such that x \ l ( x ) and R / l ( x ) i s a prime 34* semi-simple r i n g . Let 0 4 x e R . Since ( x ) R \ 34* there i s a f i n i t e l y generated subring R' C ( X ) r such t h a t R' 34* . Let Z(x) = (I<TR : R'/R' fl I | 3i*} . Let J Q : a e A be an ascending chain i n Z(x) and l e t J = U(J : a e A] . I f - 1 1 6 -J £ Z(x) then R ' / R ' fl J e 34* < 3. ft .* so R ' / R ' n J e 3. a . Now by 2 . 2 . 6 - R ' .n J i s f i n i t e l y generated as a r i n g so R' fl J c R ' n J a f o r some a e A . Thus R ' - fl J = R ' fl J which Is a c o n t r a d i c t i o n since a J a e z(x) . Therefore J e Z(x) so by Zorn's Lemma we may choose I ( x ) maximal i n Z(x) . F i r s t we s h a l l prove that R / I ( x ) i s 34* semi-simple. Let K O R such that K p l ( x ) . Then R ' / R ' f l K e 34* . Now, R , n K = ( R ' n K ) / ( R n i ( x ) ) ' T h u s l f R ' R n i ( x ) e * * T H E N R ' n i ( x ) € M * ' T h e r e f o r e  R ' n K _ R ' n K + i ( x ) i , « * • +. i pri—p j ( x ) = l ( x ) — * since 34* i s s t r o n g l y h e r e d i t a r y , K / l ( x ) i 34* . Hence R / I ( x ) i s 34* semi-simple In order to prove that R / I ( x ) i s prime we begin by showing that i f K and H are i d e a l s of R such that K p I ( x ) and H p l ( x ) then K 0 H p l ( x ) . Suppose t h i s i s not t r u e ; that i s , suppose K fl H = l ( x ) . Then the r i n g R ' n K R ' n K „ R ' n K + R ' n H . . . » R ' n i ( x ) = R ' n H n K = R ' n H 1 3 a s w b r i n s o f R ^ H T H 6 S O R | R n i ( x ) 6 w * • N o w R V ( R ' n i ( x ) ) „ R ' V * . R ' • ( R J n K ) / ( R ' n i ( x ) ) = R J n K E * S O R ' n i ( x ) £ ** ' This i s a c o n t r a d i c t i o n . Therefore K fl I p l ( x ) . - 117 -Now we can.prove that R / I ( x ) i s a prime r i n g . Suppose th a t K and H are as above and that KH c l ( x ) . Then K fl H ^ l ( x ) and (K fl H ) 2 c l ( x ) . Then / 3 ( R / I ( X ) ) 4 (°) • This i s a c o n t r a d i c t i o n since R / l ( x ) i s 34* semi-simple and jS <_ 34* . Therefore R / l ( x ) Is prime. Since R'/R' 0 I ( x ) k 34* , R' £ l ( x ) so x § I ( x ) . This completes the proof. Q.E.D. In view of 1 . 1 . 6 t h i s theorem i m p l i e s that a l l l o c a l r a d i c a l c l a s s es 34* which are between 0 and $.&.* are s p e c i a l r a d i c a l c l a s s e s . I n p a r t i c u l a r , .* i s a s p e c i a l r a d i c a l c l a s s . Theorem provides an a l t e r n a t e proof f o r Theorem 52 (£ i s a s p e c i a l r a d i c a l c l a s s ) i n D i v i n s k y [ 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 ge n e r a l i z e d and elementary c l a s s e s r e l a t e d to £ and 2s. £).*. By 2 . 4 . 2 ( v i i ) £ = £ and ( 3 . B . * ) = ( 3 . S . * ) g g-^  S g-^  We have al r e a d y seen that £ j = ]/} and <£' = VI 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 . i f and only i f there are i n t e g e r s a^, a n _ ] L such that x n + a_ , x n _ 1 + ... + a,x = 0 i t i s c l e a r that - 118 -( 3 . f c . * ) g = ( & R ) g and (ff.S . * ) ' = & R = • 5.2 THE LOCAL RADICAL CLASSES *, £) * AND F l * . Let F l be the c l a s s of a l l f i n i t e r i n g s . We s h a l l begin 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 of those r i n g s R such that every f i n i t e l y generated subring of R i s f i n i t e . I n the f o l l o w i n g p r o p o s i t i o n we c o l l e c t s e v e r a l elementary 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 r a d i c a l c l a s s . (2) £ F l * £ 3 . & . * . (3) £' £ F l * £ FC . (4) F I * = 3 . £>.* n FC . (5) £) and F I * are u n r e l a t e d . (6) FF and F I * are u n r e l a t e d . Proof: (1) C l e a r l y the con d i t i o n s of Theorem 2.2.7 are s a t i s f i e d so F I * i s a l o c a l r a d i c a l c l a s s . (2) Since <J = j3 fl FC any f i n i t e l y generated 3 r i n g must be f i n i t e and of course any f i n i t e r i n g i s i n t F . D . Therefore, J < FI <_ 3 . & .* . Since the r i n g of i n -tegers i s i n 3.S> .* , F I * £ 5 . £ .* . The r i n g F p e F I * but F <t J so <f £ F I . - 119 -(3) . By 3 . 4.4 £' <_ F l and since 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' £ F l . C l e a r l y P I * £ FC . (4) This i s c l e a r s i n c e a r i n g i n 3. £ . fl PC must be f i n i t e . (5) Since F p e P I * , P I * £ . Example 6 i n Rings and  Rad i c a l s D i v i n s k y [ 7 ] shoxtfs that the zero r i n g on the • a d d i t i v e group of r a t i o n a l numbers i s i n £) so £) i P I * . (6) Since f i n i t e f i e l d s are i n P I * , F l * £ FF . C l e a r l y FF £ P I * since Q < FF and £) £ F l * . Q.E.D. The f o l l o w i n g theorem provides an i n t e r e s t i n g c h a r a c t e r i z a t i o n of P I * . 5 . 2 . 2 THEOREM: R e P I * i f and only i f every f i n i t e l y generated subring of R s a t i s f i e s D.C.C. on l e f t i d e a l s . Proof: Since 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 c l e a r . Conversely, assume that every f i n i t e l y generated subring of the r i n g R s a t i s f i e s D.C.C. on l e f t i d e a l s . Let R' be a f i n i t e l y generated subring of R and - 120 -l e t N' = 7 l ( R / ) • The r i n g R'/N' i s Isomorphic to a f i n i t e d i r e c t sum of matrix r i n g s over d i v i s i o n r i n g s . Let R'/N' ~ (D n ) Q •• . O (A ) ' • Then each D. i s f i n i t e l y 1 n^ w k 1 generated as a r i n g and i f A^ i s a f i n i t e l y generated sub-r i n g of D. then A. s a t i s f i e s D.C.C. Let x. e D. . Then <x^>n = <x^->n+^ f o r some i n t e g e r n >. 1 so n n+1 L « . , x. = a ^ - x . + ... + a.x. f o r some i n t e g e r s a .... a T . l n+1 l i i ° n+1' ' L Since D^  has no proper d i v i s o r s of zero x ± = . a n + 1 x 2 + ... + a L x ^ " n + 1 . Therefore D± e e' so by 3 . 4 . 4 , D ± i s a f i n i t e f i e l d . Thus R'/N' I s f i n i t e and so by 2 . 2 . 6 N' i s f i n i t e l y generated as a r i n g . Since N' s a t i s f i e s D.C.C.on l e f t i d e a l s N' e J* by Lemma 28 i n Di v i n s k y [ 7 ] . By 3 . 2 . 1 J = j3 n PC so N' i s f i n i t e l y generated, 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 . Thus, N' i s f i n i t e so R' must be f i n i t e . Therefore R e P I * . Q.E.D. We now turn to a c o n s i d e r a t i o n of the l o c a l c l a s s es j f * and g>* . By 4 . 3 . 4 j>* = £* . 5 . 2 . 3 PROPOSITION: j * = £ n FC = z n PI* = y> n F I * - 121 -Proof: Since J < PC and J < X , J * < ^ n PC . I f R e £ fl PC then every f i n i t e l y generated subring of R i s 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 R e J * . Therefore, J * = £ 0 PC . Since P I * < PC , £ P. P I * < £ 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 i s f i n i t e so J * < F l * . Thus J > * = £ n F l * . As above J * < y^fl F l * . Since a f i n i t e n i l r i n g i s n i l p o t e n t and i n PC , "ft fl P I * <_ J * * . This completes the proof-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 given by Baer [ 6 ] . We present t h i s example to show that £ fl FC £ 0 and hence that J* £ jB . 5.2.4 EXAMPLE: For each i n t e g e r k l e t = (0,a(k)} =.the a d d i t i v e group of 2 elements. CO CO For each i n t e g e r i > 1 l e t T. e HomfS G, . EG,) o _ x v k' k' — 00 —CO such that , 0 i f k s 0 mod 2 1 . T ( a ( k ) ) o { x a ( k - l ) i f k £ 0 mod 2 1 . - 122 -CO CO Let R be the subring of Hom( S G k , £ Gfc) which -OS -09 i s generated by the set (T\ : i >_ 1} . (1) F i r s t we s h a l l prove that /3(R) = (0) . Let (0) 4 I <! R and l e t 0 4 X e I . Let h 2 h be any i n t e g e r >_ 1 . We s h a l l prove that I / (0) . Now X i s a sum of monomials i n the T^'s so we may w r i t e X = + .... + where I <_ k , 4 0 , V, 4 0 and V. i s a sum of monomials of len g t h i+1 . K 1 Let V, = T ... T +...+T ...T k m , - . m, m ~ m , 1,0 l , k n,0 n,k Since V k 4 0 there i s an i n t e g e r t such that V k ( a ( t ) ) 4 0 . (We may choose t > 0 sinc e t = 0 mod 2 1 i f and only i f - t s 0 mod 2 1 ) . Choose i n t e g e r s r and s such that k+1 < 2 r and m. .'_<_ r f o r a l l i and j such th a t 1 <_ i <_ n and 0 < j < k , and 2 r + 1 - 2 h + t < 2 s . r h We wish to show that ( T 2 ~ k _ 1 . X ) 2 ( a ( t + 2 h - 2 r ) ) ^ 0 , r s To begin we consider ( T 2 " k - 1 V,') (a(t + 2 h - 2 r ) ) . Consider V k ( a ( t + cp-2r)_) where cp i s an i n t e g e r • such that 1 < cp <_ 2 h . Suppose t + cp2r - j = 0 m* k- " mod 2 ± } J f o r some i and j such that 1 <_ i <_ n and 0 <_ j <_ k . Since r >_ m. , . , cp2r = 0 . m. , . i ^ - J m_ _ mod 2 1 ' K " J .. Thus t - j & 0 mod 2 X>K~2 . This i m p l i e s that T ... T maps a ( t + cp.2r) onto m. ^  m. , ^ v ^ 1 i , 0 i , k - 123 -0 i f and only i f i t maps a ( t ) onto 0 . Since V ^ ( a ( t ) ) 4 0 an odd number of the monomials must map a ( t ) onto a ( t - k - l ) so V k ( a ( t 4-cp-2 r)) = a ( t + c p - 2 r - k - l ) . 2^—k—1 r Let us now consider T ( a ( t + cp-2 - k - 1)). s Suppose I i s an i n t e g e r such that 1- <_ I <_ 2 r - k - 1. Now t + cp-2r < 2 s so t + cp-2r - k - 1 < 2 s . Since 1 <_ 2 r k - 1 , k + 1 + I < 2 r so t + c p - 2 r - k - 1 - I >_ t + cp-2r - 2 r = t + (cp - l ) 2 r > 0. Therefore, 0 < t + c p - 2 r - k - l - - f , < 2 S so T 2 ~ k _ 1 ( a ( t + cp.2r - k - 1)) 4 0 so T 2 " k " 1 ( a ( t + cp-2r - k - 1)) = a ( t + cp«2r - k - l - ( 2 r - k - 1}) = a ( t + (cp - l ) 2 r ) . I t f o l l o w s that ( T 2 r - k - l # V k ) 2 h ( a ( t + 2 h m 2 v ) ) = a ( t ) ^ 0 Now i f I = k then X = so 0 4 ^ f ' k ~ \ f h e I ^ • I f £ < k , (T2*.X)2* = Z + ( T 2 r ~ k ~ ^ • V, ) 2 h ' s 7 v s k y where Z ( a ( t + 2 h.2 r)) i s e i t h e r 0 or a sum of a ( v i)'s where each y i > t . (Since the length of z monomial appearing i n Z i s s t r i c t l y l e s s than - 124 -((k + 1) + 2 r - k - 1)2 = 2 r2 they cannot "move" a ( t + 2 h-2 r) as " f a r down" as a ( t ) ) . Therefore 0 / ( T 2 ~ k _ 1 - X ) 2 e l 2 . So i n any 2 h case I 4 (0) . Therefore no non-zero i d e a l of R i s n i l p o t e n t so j3(R) = (0) . (2) We s h a l l now prove that R e £ . Let R' = <X1,...,X > m be a f i n i t e l y generated subring of R . Let m = max{s : T occurs i n some X.} . Choose h > 2 Now i f Z e ( R ' ) h then each monomial i n Z i s of length at l e a s t h . Since h > 2 m , 2 m must d i v i d e one of k , k-1, k-h f o r a l l i n t e g e r s k . There-f o r e Z(a(k)) = 0 f o r a l l i n t e g e r s k so Z = 0 . Thus ( R ' ) h = (0) . (3) Therefore R e £ fl FC (since 2R = (0)) so R i s j8 semi-simple and i n £ fl FC . Since £ fl PC = J * , R e $ * . Therefore * £ /3 From P r o p o s i t i o n 5.2.1 (4) we know that £ F I * so s i n c e J * <_ P I * , £ J * . Combining t h i s w i t h 5.2 .4 we see that and J * * are - u n r e l a t e d . The classes J* and j8 are a l s o u n r e l a t e d since i t i s c l e a r that P i J * • The r e l a t i o n s between these r a d i c a l s classes 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. - 1 2 5 -ILLUSTRATION 3 - 126 -Let R be a non-zero J** semi-simple r i n g with D . C . C . on l e f t i d e a l s . Then R/y)(R) i s a f i n i t e d i r e c t sum of matrix r i n g s over d i v i s i o n r i n g s . Since R s a t i s f i e s D . C . C . on l e f t Ideals £ ( R ) = y | ( R ) . Thus 7\{R) fl F C(R) = £(R) n F C(R) = Jp*(R) = (0) so F C ( R ' ) i s isomorphic to an i d e a l of R / Y l ( R ) and hence i s a f i n i t e d i r e c t sum of matrix r i n g s over d i v i s i o n r i n g s of f i n i t e c h a r a c t e r i s t i c . Suppose that x e R such that 0 / x e F C ( R / y 2 ( R ) ) , then the i d e a l generated by x contains an i d e n t i t y e and e e F C ( R / 7 f ( R ) ) . Therefore ne = 0 f o r some i n t e g e r n / 0 so ( n e ) k = 0 f o r some p o s i t i v e i n t e g e r k , t h i s i m p l i e s that e k e F C ( R ) so e = e k e ( F C ( R ) + 7 7 ( R))/y ? ( R ) . Therefore, FC(R/YI(R)) = ( F C ( R ) + y i(R ) )/n(R) • Now J <_ * so by Lemma 28 i n D i v i n s k y [7] i f R e Yl t h e n R e J* - Since R <t J> * , ")/|(R) / R . This completes the proof of the f o l l o w i n g theorem. 5 . 2 . 5 THEOREM: D.C.C. on l e f t i d e a l s then R k Yl , FC(R) i s a f i n i t e d i r e c t sum of matrix r i n g s over d i v i s i o n r i n g s of 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 matrix r i n g s over d i v i s i o n r i n g s of c h a r a c t e r i s t i c 0 . I f (0) 4 R i s a J>* semi-simple r i n g w i t h Notice that i f R i s not only semi-simple - 127 -but a l s o P I * semi-simple then a l l the d i v i s i o n r i n g s are i n f i n i t e . I f ( R / 7 J ( R ) ) e PC then J}(R) i s a d i r e c t summand of R ; i n f a c t , R = Y|(R) ©PC(R) . Prom Theorem 13 i n D i v i n s k y [7] we conclude that ^ * ( R ) may not be equal to Yl(B.) since £ $ * . However, i f R e PC then J > * ( R ) = 7 l ( R ) since J * ( R ) = X(R) n FC(R) = £ ( R ) = 71(H) • Ot course, as we no t i c e d above, R e ^f* i f and only i f R e ) | Suppose now that R i s a non-zero r i n g w i t h A.C.C. on l e f t i d e a l s . Then /3(R) = £ ( R ) = 7 i ( R ) so jf(R) = J*(R) = $'{R) since J = /3 fl PC , J** = <£ n PC and 1 ' = T in PC . Unfortun a t e l y we cannot use Goldie's Theorem to obt a i n a r e s u l t s i m i l a r to 5 .2.5. F i r s t of a l l , I f R i s <§ * semi-simple, R may be i n 0 ( f o r example, Cro e j3 but Cm i s j f * semi-simple). Even i f R $ /3 , FC ( R/j8 ( R ) ) may not be the same as (FC ( R ) + /3(R))//3(R) . To see t h i s consider the r i n g R = ( p [ X ] / ( 2 X 2) . Then FC ( R ) = (X 2). and YXR) = J8(R) = (2X) R . C l e a r l y FC ( R ) D Y|(R) = (0) and R s a t i s f i e s A.C.C. but R / y f(R) e PC . However, i f R e PC then c l e a r l y j3 ( R ) = J*(R) . A r i n g <x> i s f i n i t e I f and only i f <x> e %• 0 PC so i t f o l l o w s that ( P I * ) ' = L n FC and - 128 -(PI*) = (FI* ) = (&„) n FC . Since J * = £ n PC i t i s v y g v 'gjL R ; g c l e a r the ( J * ) ' = 7}n PC and (J*)v = H FC . a o From 2.1 . 3 ( i i i ) , < . J ' and c l e a r l y J" < So- < p c s o o f course J** = £ n ^ ' = £ n «f . v g s The r e l a t i o n s h i p s between these r a d i c a l classes 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 -5-3 LOCAL RADICAL CLASSES tt FOR WHICH Z 4 tt 4 71 . •We have already seen that 6* = <£* = Z . For any c l a s s tt , tt* <_ tt' so from Theorem 3 . 3 - 2 we may conclude that tt* = y? f o r a l l classes of r i n g s tt such that YJ <_ tt <_ FF . Since Yl < Yl and FF <_ FF i t f o l l o w s s s that ( t t ) * = (tt )* = (tt )' = (tt )' =7? f o r a l l classes e> e>^  & &2_ of r i n g s tt such that 7? 1 M 1 F F • R e c a l l that 1£ i s the upper r a d i c a l c l a s s d e t e r -mined by the c l a s s of a l l simple idempotent r i n g s . C l e a r l y 0 <U<F s o VJ = YL - Therefore yj* <_ = YL - To prove that t l * = YL i t i s s u f f i c i e n t to show that a f i n i t e -l y generated n i l r i n g cannot be homomorphically mapped onto a simple 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 generated n i l r i n g i s not idempotent. Proof: Let (0) 4 R = <x 1, .•. ., x n> be a f i n i t e l y generated - n i l r i n g . 2 Suppose that R = R == Rx, + ... + R r . Choose n a minimal subset fx.. , x. } of ( x , , x } such x l 1 k i n that R = Rx. + ... + Rx. 1 1 x k - 130 -Since x^ e R there are elements v\> ' ' ' >v\t e R such that x. = r n x . +-...+ r, x. (*) x l 1 X l K 1 k Since R i s n i l r , m - x . '= O ' X . ' = 0 e Rx. + ... + Rx. f o r some i n t e g e r m >_ 1 . Let t be the sm a l l e s t i n t e g e r P >_ 1 such that x, x. e Rx. + ... + Rx. . Then from (*), 1 1 1 1 2 x k •t—1 f f -1 •£ —1 r , x. = r"x. + r,' r„x. + ... + r,' r, x. e Rx. + ... -i- Rx 1 1 i 1 1 2 i 2 1 k i k i 2 Since -t i s minimal, -t = 1 so x. e Rx. + ... + Rx. X l X2 Xk This c o n t r a d i c t s the m i n i m a l i t y of k . Since we have reached a c o n t r a d i c t i o n we may 2 conclude that 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, R' cannot be homomorphically mapped onto a non-zero idem-potent r i n g then c e r t a i n l y R e FF' = Yi . I t f o l l o w s then that R e Yl i f and only i f no f i n i t e l y generated subring of R can be homomorphically mapped onto a non-zero idempotent r i n g . 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 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 r i n g s with idempotent he a r t s . The f o l l o w i n g lemma w i l l enable us to - 131 -.0 prove that jS * i s a r a d i c a l c l a s s . 5 . 3 . 2 LEMMA: I f S i s a non-zero simple 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 heart. Proof: Let (0) / S = S be a simple r i n g . By Theorem . 55 i n D i v i n s k y [7] S ^ £ and so by Theorem 53 i n D i v i n s k y [7] there i s a • x e S such that x / 0 . Then (0) / Sx 2S <f S so S = Sx 2S . Thus there are elements r l J ' '"' r k a n c i s l > ''"> s k ^ n S such that k 2 x = S r.x s. . (*) i = l 1 1 Let S' be the subring of S which i s generated by the set ( x , r ^ , T\^'si> • • • >S\<Z ' Choose I maximal i n Z = ( J < s' : x £ J} . Then s'/I i s s u b d i r e c t l y I r r e -d u c i b l e w i t h heart H = ((x) + I ) / I . I f x 2 e I then by (*) x e I so - x 2 £ I . Therefore H 2 / (0) so H 2 = H . Q.E.D. An I n t e r e s t i n g conclusion that f o l l o w s from t h i s lemma i s that i f there i s a simple idempotent n i l r i n g then there i s a simple idempotent n i l r i n g which i s the heart of - 132 -a f i n i t e l y generated n i l ring. 5 . 3 . 3 PROPOSITION: (1) 8 * < 0 and so 8 * i s a r a d i c a l c l a s s . (2) £ < 6^* < Y l • (3) B^* 4 Yl i f and only i f there i s a non-zero simple . idempotent n i l r i n g . Proof: (1) Let R e fi. * . I f R I |3 then R can be homomor-p h i c a l l y mapped onto a subdirectly i r r e d u c i b l e r i n g with a simple Idempotent heart S . But then by 5 - 3 . 2 some f i n i t e l y generated subring of R i s not i n B^ . This i s a contradiction so R e S . By Theorem 2 . 2 . 2 8 * c^p J c^p i s a l o c a l r a d i c a l c l a s s . (2) By Theorem 55 i n Divinsky [7] no simple idempotent r i n g i s i n £ . Therefore £ < 8 * . Clearly S < P so — 1 cp cp — (3) I f 6 * £ 71 then there i s a n i l r i n g which can be homomorphically mapped onto a subdirectly i r r e d u c i b l e . r i n g with an idempotent hear H 4 (0) . Clearly H i s a simple idempotent n i l r i n g . Conversely, any simple idempotent n i l r i n g S / (0) i s 8^ semi-simple (and hence 8^* semi-simple since 8^* <_ B ) but i s i n Yl . Q.E.D. - 133 -In view of 1.1.6 the f o l l o w i n g theorem Implies that jS^* i s a s p e c i a l r a d i c a l c l a s s . 5.3.4 THEOREM: • A r i n g R i s /3^* . semi-simple i f and only i f R i s isomorphic to a s u b d i r e c t sum of prime 0^* semi-simple r i n g s . Proof: Since s u b d i r e c t sums of semi-simple r i n g s are semi-simple one d i r e c t i o n i s c l e a r . Conversely, l e t R be a j3 * semi-simple r i n g . I t Cp Is s u f f i c i e n t to prove that f o r a l l non-zero x e R there i s an i d e a l l ( x ) such that x l ( x ) and R/I(x) i s a prime 0^* semi-simple r i n g . Let 0 / x e R . Since ( X ) R £ i 3 ^ * there i s a f i n i t e l y generated subring R' of ( X ) R and an I d e a l l ' of R ' such that R ' / l ' contains a non-zero simple idempo-tent heart S ' / l ' . Notice that since S ' / l ' i s simple i f J < R and I ' + (R' n J) | S ' then R ' n J c i ' . (*) Let Z = [ J <? R : i ' + (R ' n J) ± S'} . Let J : a € A be an ascending chain of i d e a l s i n Z and l e t J — U [ J : a e A} . By (*) , R' fl J c I ' f o r a l l a e A . Then R ' n J c I' hence I ' + R' P, J = I ' £ S' so J e Z . Therefore by Zorn's Lemma we may choose I ( x ) ' maximal i n Z. - 134 -F i r s t we s h a l l prove that R / I ( x ) i s 8 * semi-cp simple. Let L / I ( x ) be a non-zero i d e a l of R / I ( x ) . Since L ^ l ( x ) , i ' + R 7 n L o S 7 . Now R 7 0 L + l(x) „ R 7 fl L „ . R 7 D L . . — i ( x ) = R ' n i ( x ) a n d R ' n i ( x ) c a n b e h o m o m o r -p h i c a l l y mapped onto - — ^ j ^ " + s i n c e by (*) p' n T / V \ r- T ' S ' r- R / H L + 1 7 „ , _ R 7 fl + L -f I ' K il l ( x ) c i . Now j r c j-? and so i s s u b d i r e c t l y i r r e d u c i b l e w i t h idempotent heart S 7 / I 7 . Then ( R 7 fl L + 1 7 )/1 7 i s not i n 8 so since 8 * < 8 , (R' n L + l ' ) / l ' . Thus R / I ^ X ) and hence L / I ( x ) i s not i n 3^* . Therefore R / I ( x ) i s 8^* semi-simple. Now we s h a l l prove that R / I ( x ) i s a prime r i n g . Suppose that L / l ( x ) and H/l(x) are non-zero i d e a l s of R / I ( x ) and LH c l ( x ) . By the maximal!ty of l ( x ) , I ' + ( R 7 D L) 3 S 7 and I 7 + ( R 7 fl H) 3 S 7 . Therefore s ' s / 2 + i 7 _ ( R ' n L + i U R 7 n H + i 7 ) _ i ( x ) n R 7 This i m p l i e s that S 7 c= I 7 + ( l ( x ) D R 7 ) which i s a c o n t r a -d i c t i o n since l ( x ) e Z . Therefore R / l ( x ) i s a prime 8^* semi-simple r i n g and since R 7 £ l ( x ) , x { l ( x ) . This completes the proof. Q.E.D. - 135 -5 . 4 LOCAL COMPLEMENTARY RADICAL CLASSES. Let d be a r a d i c a l c l a s s . I f there i s a r a d i c a l c l a s s ft such t h a t : (1) ft(A) n $ (A) = (0) f o r a l l r i n g s A . (2) I f 3" i s a r a d i c a l c l a s s such that 3(A) n J (A) = (0) f o r a l l r i n g s A then 3" <_ ft . then Andrunakievic [2] defines ft to be the complement of J . We s h a l l denote ft by CRK(q£ ) . Notice t h a t f o r some r a d i c a l c l a s s e s J , CRH(^) may not e x i s t . In [2] Andrunakievic proves the f o l l o w i n g theorem. 5 . 4 . 1 THEOREM: I f ft i s a h e r e d i t a r y 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) = the upper r a d i c a l c l a s s determined by the c l a s s 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 r i n g s w i t h hearts i n M . ( i i ) R e CRH(ft) i f and only i f every homomorphic image of R i s ft semi-simple (such r i n g s are c a l l e d s t r o n g l y ft semi-simple). 5 . 4 . 2 DEFINITION: Let be a r a d i c a l - c l a s s . I f there i s a l o c a l r a d i c a l c l a s s ft such t h a t : ( i ) ft(A) fl J (A) = (0) f o r a l l r i n g s A . - 136 -( i i ) I f 3 i s a l o c a l r a d i c a l c l a s s and 3(A) n jtf (A) = (0) f o r a l l r i n g s A then 3 <_ M . then tt i s the l o c a l complement of We w i l l denote the l o c a l complement of j$ by $ . 5 . 4 . 3 THEOREM: I f tt* i s a l o c a l r a d i c a l c l a s s then tt* e x i s t s and tt* = CRH(Ji*)* . Proof: Notice that tt* i s h e r e d i t a r y so CRH(W*) e x i s t s . We s h a l l prove that CRH(W*)* i s a r a d i c a l c l a s s which s a t i s f i e s c o n d i t i o n s ( i ) and ( i i ) of 5-4.2. (1) Since CRH(tt*) s a t i s f i e s c o n d i t i o n (A) so does CRH(W*)*, Suppose that B i s an i d e a l of a r i n g A and that both A/B and B are i n CRH(W*)*'. Let A' be a f i n i t e l y generated subring of A . I f A' i s not s t r o n g l y tt* semi-simple then A can be homomorphically mapped onto a non-zero r i n g which i s not tt* semi-simple. Thus there i s a f i n i t e l y generated subr i n g L' of A' Such that L' can be homomorphically mapped onto (0) / (L'/K') e tt* . Since A/B e CRH(M*)* , L B e CRH(tt*) . Thus ^ + B ~ ^ ± s s t r o n g l y tt* semi-simple. I t f o l l o w s that (L' ( I B ) + K' = L ' (since L ' / L ' fl B can be homomorphically mapped onto - 137 -L'/(L' fl B + K )) . Therefore L' fl B can be homomor p h i c a l l y mapped onto - — n K ? + K = L'/K' e ft* . This Is a c o n t r a d i c t i o n since i / fl B'c B and hence i s i n CRH(ft*) * (so no f i n i t e l y generated subring of L ' n B can be homomorphically mapped onto a non-zero r i n g i n ft*) . Therefore every f i n i t e l y generated subring of A i s s t r o n g l y ft* semi-simple so by 5-4.1 ( i i ) A e CRH(ft*)* . Then by P r o p o s i t i o n 2.2.1 CRH(ft*)* i s a l o c a l r a d i c a l c l a s s . (2) Let A be a r i n g and l e t I = ft* ( A ) fl CRH(ft*)*(A) . Let R' be a f i n i t e l y generated subring of I . Then R' e ft* and R' e CRH(ft*) so R' = ft*(R') fl CRH(W*)(R /). Therefore R' = (0) since CRH(1A*) i s the compliment of ft* . Hence I = (0) so c o n d i t i o n ( i ) of 5.4.2 i s^ s a t i s f i e d . (3) Suppose that 3" i s a l o c a l r a d i c a l c l a s s and 3(A) n ft*(A) = (0) f o r a l l r i n g s A . Then 3 < CRH(ft*) si n c e CRH(ft*) i s the compliment of ft* . But then 3 = 3* <_ CRH(ft*)* so c o n d i t i o n ( i i ) of 5.4.2 i s s a t i s f i e d . Therefore ft* = CRH(ft*)* . Q.E.D. I f f o l l o w s that i f ft* i s a l o c a l r a d i c a l c l a s s then R e f t * i f and only i f every f i n i t e l y generated subring - 138 -of R i s s t r o n g l y tt* semi-simple. The f o l l o w i n g theorem shows that there i s no need to define elementary complements of l o c a l r a d i c a l c l a s s e s . 5 - 4 . 4 THEOREM: I f tt* i s a l o c a l r a d i c a l c l a s s then ( t t * ) 7 = tt*. Proof: Let tt* be a l o c a l r a d i c a l c l a s s . Since tt* i s a l o c a l c l a s s , (tt*) <_ ( t t * ) 7 . Let R e (W*)' and l e t R 7 be a f i n i t e l y genera-ted subring of R . Let R 7 / l ' be a homomorphic image of R' and l e t J 7 / I 7 = t t * ( R 7 / I 7 ) . I f x e J 7 / l ' then <x> e tt* and since R e ( t t * ) 7 <x> € tt* . Thus <x> = (0) so J ' / l ' = (0) . Therefore, R 7 i s s t r o n g l y tt* semi-simple so R e t t * . Hence, tt* = (M*) 7 i s an elementary c l a s s . Q.E.D. Notice that i f tt > ft then CRH(tt) <_ CRH(ft) i f they both e x i s t and tt <_ ft i f both 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 be a r i n g which i s not i n tt* . Then some f i n i t e l y generated sub-r i n g R 7 of R i s not s t r o n g l y tt* semi-simple. Thus R 7 can be homomorphically mapped onto a r i n g R" such that - 139 -tt*(R") 4 (0) . C l e a r l y then M*(R*) 4 R" since tt*(R ) fl tt*(R ) = (0), . Therefore, R | tt* . I t f o l l o w s that tt* <_ f * . Suppose that ft <_ tt <_ 3" and a l l three classes are r a d i c a l c l a s s e s . Also assume that 3 and ft e x i s t and that f = ft . Then 1T(R) fl tt(R) c T T ( R ) n 3T(R) = (0) f o r a l l r i n g s R . 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 R then J (R) fl ft(R) c ^ ( R ) n tt(R) = (0) f o r a l l r i n g s R so J <_ ft = If . Therefore 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 the l o c a l complements of the r a d i c a l classes we have been d i s c u s s i n g . 5.4.5 PROPOSITION: P I * = {(0)} Proof: We need only show that i f tt* i s a l o c a l r a d i c a l c l a s s and tt* 4 ( ( 0 ) } then PI* ( A ) n tt*(A) 4 (0) f o r some r i n g A . Suppose that 0 4 R e tt* and that tt* i s a l o c a l r a d i c a l c l a s s . Let 0 / x e R . Then <x> e tt* and so i s 2 2 ? <x>/<x> . I f <x> 4 <*> then <x>/<x> can be homomor-p p h i c a l l y mapped onto a f i n i t e r i n g . I f <x> = <x> f o r a l l non-zero x e R , then R e fi' so every f i n i t e l y generated - 140 -subring i s f i n i t e . In e i t h e r case we see that there i s a f i n i t e r i n g i n tt* . This completes the proof. Q.E.D. Of course the class. ((0)} i s the l o c a l comple-ment of the c l a s s of a l l r i n g s . I t then f o l l o w s from 5 - 4 . 5 that i f tt i s a r a d i c a l c l a s s and tt >_ P I * then tt e x i s t s and tt = {(0)} . In p a r t i c u l a r then, FT* = PC = W7&7* = ~&R = T = ( £ R ) G =•"£ G = ((0)} . 5 . 4 . 6 PROPOSITION: A , P and FF e x i s t and J = P = PP = . 0 g g Proof: Let R be a r i n g and l e t I = e'(R) n FF (R) . I f 1/ (0) then since I e £' by 3 - 4 - 3 I can be homomor-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 K of prime c h a r a c t e r i s t i c . Since I «3 FP (R) , FF (R) can be homomor-g g p h i c a l l y mapped onto K (see Theorem 4 6 i n D i v i n s k y [ 7 ] ) . Since a l l f i n i t e l y generated subrings of K are f i n i t e f i e l d s t h i s i s a c o n t r a d i c t i o n . Therefore I = (0) so £ ' fl PP = ((0)} . S i m i l a r i l y one shows that e' n P = {(0)}. Since J < PF_ , £' H J = ((0)} . Suppose th 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 -2 34 4. £' . Then there Is a r i n g <x> e 34 such that <x> ¥ <*> < Thus <x> can be homomorphically mapped onto the t r i v i a l r i n g 2 <x>/<x> and so <x> can be homomorphically mapped onto a simple zero r i n g . Thus 34 fl J / { ( 0 ) } . I t f o l l o w s that 34 n PF -/ ( ( 0 ) } and 34 fl F 4 C(0)} -Therefore, £' = J = F =. FF f Q.E.D. Now i f $$ i s any r a d i c a l c l a s s such that Of <_ and & < F or. ^ <_ PF then = £' . This i n c l u d e s a l l of the r a d i c a l c l a s s es l i s t e d i n Chapter I except FF ( f o r example : J = 71 = £'). I t a l s o includes the ge n e r a l i z e d l o c a l and elementary classes determined by these r a d i c a l c l a s s e s ( f o r example : J = J = £') and the l o c a l and g g-j_ . — elementary c l a s s e s determined by these r a d i c a l c l a s s e s ( f o r example : J * = J ' = £'). Since any r i n g R can be embedded (as an i d e a l ) . i n a r i n g R, wit h i d e n t i t y i t i s c l e a r t h a t £ = { ( 0 ) } . So 1 s l of course, i f <$ i s any r a d i c a l c l a s s and J >_ £ g then T = ((on . The r a d i c a l c l a s s FP does not have a l o c a l com-plement; that i s , PF does not e x i s t . To see t h i s consider the r a d i c a l classes ^n = ® ^ p ^ S n ^ : P i s a Prime] - 142 -where i s the set of a l l p o s i t i v e i n t e g e r s <_ n . Let R be a r i n g and l e t I = ^ n ( R ) n • Since I e 3" <. £' , i f I 4 (0) , I can be homomorphically mapped onto a f i e l d K i n £' (see 3«4.3)« However, since I e 3 , K e 3 , so K i s a f i n i t e f i e l d . But then (as i n n 3 n ' v Theorem 46 i n Div i n s k y [7]) FF(R) can be homomorphically mapped onto K . This Is a c o n t r a d i c t i o n because K i s f i n i t e . Therefore I = (0) so 3^ fl PP = ((0)} . I t f o l l o w s that i f PP e x i s t s then Z c PP f o r n — a l l p o s i t i v e i n t e g e r s n . But then U 3^ c FF so £ ' = (u 3 )' c (FF) ' = PP . This i s impossible since n n £' fl PP contains a l l i n f i n i t e a l g e b r a i c f i e l d s of f i n i t e c h a r a c t e r i s t i c . Therefore, PP does not e x i s t . We s h a l l now consider the l o c a l complements of elementary r a d i c a l s which are <_ PC . 5.4.7 PROPOSITION: (1) R e PC i f and only I f f o r a l l x e R , 2 k a l x + a 2 x + ••' + a^x = 0 f o r some i n t e g e r s a^,...,aj such that p d i v i d e s a l l f o r i > 1 but p does not d i v i d e a^ . (2) I f S 4 gf then 3 p71(S) = FC p . - 143 -(3) R e 3:pYl (0) i f and only i f f o r a l l x e R , a-^ x + . .. + a^x^ = 0 f o r some i n t e g e r s a^, . .., a^ such that p does not d i v i d e . (4) I f S 4 0 then R e 3 (S) i f and only i f f o r a l l x e R , a^x +• ... + a^x^" = 0 f o r some i n t e g e r s a l ' a k s u c h that p does not d i v i d e a. • f o r some j but p d i v i d e s a l l a i f o r which i 4 j • (5) FC = PC . X J ! P P (6) ^n(^) • ( 7 ) T^sJ = ffp(z+) i f Proof: ( l ) Suppose R- i s a r i n g such that f o r a l l x e R , a^x + .. . + a k x k = 0 f o r some i n t e g e r s a^, . . a^ such t h a t p does not d i v i d e a^ but p d i v i d e s a^ f o r a l l i > 1 . I f FC (R) 4 (0) then there i s a x e PC (R) p ' ' P such that x / 0 but px = 0 . But there are i n t e g e r s a l ' a i c s u c n that p d i v i d e s a^ f o r i > 1 but p does not d i v i d e a^ and a-^ x + .. . + a^x k = 0 . Then a^x = 0 and since p does not d i v i d e a^ , x = 0. This i s a c o n t r a d i c t i o n so PC p(R) = (0) . I t f o l l o w s that every f i n i t e l y generated subring of R i s s t r o n g l y - 144 -PC semi-simple so R e PC P P Let R e FC and l e t x e R . Then <x> i s P s t r o n g l y FC p semi-simple so <x> = <px> . Thus there are i n t e g e r s a.^ . a^ such that p d i v i d e s a l ' '''' a k a n d x = a l X + '"' + a k x l C ' Therefore 2 k (a^ - l ) x -8- a 2x + ... + a^x = 0 and c l e a r l y p does not d i v i d e a^ - 1 . This completes the proof of ( l ) . ( 2 ) , ( 3 ) , (4) The proofs f o r (2), (3), (4) are i n a l l respects s i m i l a r to the proof of ( l ) . We have already seen that FC p <_ FC p . Suppose R £ FCp . Let R' = R/FC p(R) . Let 0 4 x e R' . I f FC(<x>) 4 (0) then there i s a y e <x> such that qy = 0 f o r some prime q / p . I f FC(<x>) = ( 0 ) c o n s i s e r 7L(<x>) . I f 7£(<x>) 4 ( 0 ) there i s a y e <x> such that <y> = 0°° so <y> can be homomorphically mapped onto C^ f o r some prime q 4 p . I f Vl(<x>) = 0 then as i n .the proof of 3 - 3 . 2 <x> can be homomorphically mapped onto a f i n i t e f i e l d of c h a r a c t e r i s t i c q 4 P • Thus In any case there i s a subring of R which i s generated by one element and which can be homomorphically mapped onto a r i n g of prime c h a r a c t e r i s t i c q 4 P • I f q Is a prime and q 4 P then a r i n g of - 145 -c h a r a c t e r i s t i c q i s i n PC (take a n = q a. = 0 p l i f o r 1 > 1). Therefore, i f R £ FC p there i s an element x e R such that <x> i s not s t r o n g l y FC p semi-simple. Thus R £ FC p . I t f o l l o w s that PC3 = FC P P • (6),(7) The proofs f o r (6) and (7) are s i m i l a r to the proof of (5). Q.E.D. In view of the f o l l o w i n g theorem P r o p o s i t i o n 5.4.7 completely determines the l o c a l complements of elementary r a d i c a l c l a s s e s which are <_ PC . 5.4.8 THEOREM: I f 34 <_ FC , 34 i s an elementary r a d i c a l c l a s s and 34 = © [tt : p e S) i s the r e p r e s e n t a t i o n of tt given i n 3-4.14 then tt = n(¥ : p e S} . Proof: Let 34 = ^ (34p : p e S} be-an'elementary r a d i c a l . Since 34 > 34 f o r a l l p e S , tt < tt f o r a l l — p ^ P p e S . Therefore tt <_ n(¥ : p e S] . Suppose R e n(34p : p e S} and x e R . Then <x> i s s t r o n g l y • 34p semi-simple f o r a l l p e S so c l e a r l y <x> i s s t r o n g l y 34 semi-simple. Thus R e 34 . - 146 -Therefore ft = n{)( : p e S} Q . E . D . By v i r t u e of 5.4.7 and 5.4.8 we can conclude: Since &' = (Z +) : p i s a prime} e' = fl(3 (z+) : p i s a prime} = Jl P Since o f ' = + f.3" : P i s a prime} J ' = n[3p~^ TpT : p i s a prime} = e' . Since & R n PC = ©(3 plt(Z +) : p i s a prime} <&R n PC = n(3: p^(Z +) : p i s a prime} = {(0)} . We have seen that ft' = ft and = e' . In f a c t , t h i s i m p l i e s t h a t : 5.4.9 PROPOSITION: For 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 only i f 34* <_ t' . (2) 34* n £' = {(0)} I f and only i f 34* < % . Proof: Let .34* be a l o c a l r a d i c a l c l a s s . Since 34* , YL and &' are h e r e d i t a r y i t f o l l o w s that i f 34* n Jl = ((0)} then ft* 1 TJT = and i f ft* fl £' = ((0)} then ft* < T = • - 147 -C l e a r l y i f 34* n | 4 ( ( 0 ) ) then 34* £ &' and i f 34* n &' 4 { ( 0 ) } then 34* £ . Q.E.D. 5 - 5 A REPRESENTATION OF &' AS THE INTERSECTION OF RADICAL CLASSES, Many of the r a d i c a l classes which we have discussed can be represented as the i n t e r s e c t i o n of two other r a d i c a l c l a s s e s . 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.* , F I * = RFC and J** = £ fl F I * . I t seems n a t u r a l to ask i f there i s a r a d i c a l c l a s s 34 £ e' such that F I * fl 34 = e' . We s h a l l see that no such c l a s s e x i s t s i f we demand tha t i t be a l o c a l c l a s s . However, - 148 -we s h a l l prove that CRH( Jl ) fl F I * = &' . g Me s h a l l begin w i t h the f o l l o w i n g easy lemmas 5 . 5.I LEMMA: I f R e F I * and R i s Y| semi-simple then g R e e' . Proof: Let R be a semi-simple r i n g which i s i n F I * Let x e R . Then <x> i s f i n i t e and /£(<x>) = (0) so <x> i s a commutative Wedderburn r i n g . Thus <x> i s a f i n i t e d i r e c t sum of f i e l d s so <x> = <x> 2 . Therefore R e &' . Q.E.D. 5.5.2 LEMMA: (CRH(7l g))* = C' = (CRH( 7?^)) ' . Proof: I f R e S' then every subring of R i s s t r o n g l y Yl semi-simple. Therefore £' <_ (CRH( Yl ))* . g g Suppose R e (CRH( 7? ) ) ' . Let 0 4 x e R . Then o <x>/<x> i s VI semi-simple so <x> = <x> . Therefore g R e &' . Therefore t' < (CRH(Yl ) ) * <_ (CRH(*y^))' < &' . Q.E.D. - 149 -Notice that i n the proof of 5 . 5 . 2 we a c t u a l l y show that i f <x> e CRH( ) then <x> e £ . Thus (CRH (71 g ) ) g i < e • 5 . 5 . 3 THEOREM: CRH( 77 ) n PI* = e' and i f ft i s a l o c a l r a d i c a l c l a s s ' such that . ft n F l * = £' then ft <_ CRH( $ ) so ft = e' . Proof: As we saw i n 5 - 5 . 2 , £' < CRH(W ) • Thus &' <. CRH( fl ) n F I * . I f R e CRH( $ ) n F I * then R i s Yl semi-simple g s so by Lemma 5-5-1* R e £' . Therefore CRH(V| ) n P I * = &' . Suppose that ft i s a l o c a l r a d i c a l c l a s s such that ft n P I * = e' . I f ft i CRH(Y| ) then there i s a r i n g R e f t such that R i s not s t r o n g l y Y| semi-simple. Since ft i s a l o c a l c l a s s there i s a r i n g <x> e ft such that <x> i s n i l p o t e n t . Then <x> can be homomorphically mapped onto a f i n i t e n i l r i n g <x'> . The r i n g <x'> e P I * D ft = e' . This i s a c o n t r a d i c t i o n so ft <_ CRH(y| ) Now ft = ft* <_ (CRH(Y| ) ) * = e ' by-Lemma 5 - 5 - 2 . Therefore ft = &' . Q.E.D. - 1 5 0 -Of course CRH(Yjg) 4 since the r i n g of r a t i o n a l numbers Q e CRH( Y| ) . This example a l s o show that CRH( V / ) i PC . On the other hand, 7J fl FC { ( 0 ) } so PC 4 CRH(Y| ) • Notice that CRH(Y? ) 0 FC ^ S' . For an example 6 consider any f i e l d of f i n i t e c h a r a c t e r i s t i c which i s not a l g e b r a i c . 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 the 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 l o c a l c l a s s e s ft* which are both r a d i c a l c l a s s e s and semi-simple c l a s s e s . We s h a l l see that a l l r a d i c a l classes which are 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 classes (indeed, elementary'classes) so we s h a l l begin w i t h the more general problem. 5 . 6 . 1 LEMMA:-I f ft i s a c l a s s of r i n g s such that s u b d i r e c t sums of r i n g s i n ft are i n ft and such that ft s a t i s f i e s condi-t i o n (A) then ft i s s t r o n g l y h e r e d i t a r y . Proof: Let ft be a c l a s s of r i n g s such that s u b d i r e c t sums of r i n g s i n ft are i n ft and such that ft s a t i s f i e s c o n d i t i o n (A). i n t e g e r s . 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 product (complete d i r e c t sum) Let R e f t and S be a subring of R . + {R± : i e Z } . A ( S ) + E(R ± : i e Z +} i s a su b d i r e c t sum of copies of R and hence i s i n ft , so - 152 -A(S) + S[R. : 1 e Z +} S ~ A(S) ~ - e ft . SCR ± : i e Z +} Q.E.D. Using a theorem of .Amitsur [ l ] which s t a t e s that every r i n g i s a homomorphic image of a su b d i r e c t sum of t o t a l matrix r i n g s of f i n i t e order over the r i n g of a l l i n t e g e r s , Armendariz i n [5] proves that i f a hy p e r n i l p o t e n t r a d i c a l c l a s s ft i s a semi-simple c l a s s , then ft contains a l l r i n g s . R e c a l l that a hyp e r n i l p o t e n t r a d i c a l c l a s s i s a he r e d i t a r y r a d i c a l c l a s s which contains a l l n i l p o t e n t r i n g s . 5 . 6 . 2 THEOREM: I f ft i s a semi-simple r a d i c a l c l a s s and ft £ &' then ft i s the. c l a s s of a l l r i n g s . Proof: Let ft be a semi-simple r a d i c a l c l a s s . I f ft ^  £' then there i s an R e f t and an x e R such that <x> 4 <x> • Since ft i s a semi-simple c l a s s s u b d i r e c t sums of r i n g s i n 2 ft are i n ft so by 5 . 6 . 1 <x> e ft . Now <x>/<x> i s a 2 zero r i n g on a c y c l i c group and <x>/<x> e ft . Since ft s a t i s f i e s ( P ) , C w e ft . Therefore 6 <_ ft since 6 = the lower r a d i c a l c l a s s determined by [Ca] . Therefore ft i s a hy p e r n i l p o t e n t r a d i c a l c l a s s so by the preceding remarks ft i s the c l a s s of a l l r i n g s . Q.E.D. - 153 -We can now prove that a l l semi-simple r a d i c a l classes must be elementary c l a s s e s . 5 . 6 . 3 LEMMA: I f ft i s a semi-simple r a d i c a l c l a s s then ft = ft'. Proof: Let ft be a semi-simple r a d i c a l c l a s s . Then by 5 - 6 . 1 , ft <_ ft' . I f ft i s the c l a s s of a l l r i n g s then c l e a r l y ft = ft . I f ft i s not the c l a s s of a l l r i n g s then by 5 . 6 . 2 , ft <_ e' so &'<&'. Suppose (0) 4 R e ft' . Then by 3 . 4 . 3 R i s isomorphic to a su b d i r e c t sum of f i e l d s F : a e A where each F„ i s an a l g e b r a i c f i e l d of prime c h a r a c t e r i s t i c . Let j3 e A . Then f o r a l l x e F^ , <x> e ft since R e f t ' . Therefore the d i r e c t product (complete d i r e c t sum) A =~j]"(<x> : x e P^} e ft <_ £' . Let y e A such that y(x) = x f o r a l l x e . By 3 . 4 . 6 <y> i s f i n i t e . Therefore F^ must be f i n i t e . Since f i n i t e f i e l d s are generated by one element each F i s i n ft and since sub-a d i r e c t sums of r i n g s i n ft are i n ft , R e ft . Hence ft' <_ ft . Therefore ft = ft' so ft i s an elementary c l a s s . Q.E.D. - 154 -5 - 6 . 4 LEMMA: I f 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 set of f i n i t e f i e l d s then a r i n g R i s isomorphic to a s u b d i r e c t sum of f i e l d s i n 3 i f and only i f every f i n i t e l y generated subring of R i s isomorphic to 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 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 set of f i n i t e f i e l d s . Then i f F e 3 there i s an i n t e g e r n(P) such that x n ^ ^ = 1 f o r a l l x e F - Let N ="["[{n(F) : F e 3} + 1 . Then I f x e F e 3 , x N = x . Assume that R has i d e a l s I : a e A such that R / I Q S P a e 3 and n ( I a : a e A} = (0) . Let R' be a f i n i t e l y generated subring of R . By 3 - 4 - 5 R' e £' . Then by 3 - 4 . 4 R 7 = A x © . . . © A k where the A ± are f i n i t e f i e l d s , Choose a. e R, such that <a. > = A. . Then a. / 0 so I i i I a. £ I_ f o r some j3. e A . Now <a.> fl Ia <1 <a.> so i 1 p\ * i I I <a.> n I = (0) . Therefore A. ~ < a, > = (<a."> + I Q )/Ia i p. l I x i p . p . i * i ' I i s Isomorphic to a subring of F s . Since 3 i s s t r o n g l y ' I h e r e d i t a r y R' i s isomorphic to a f i n i t e d i r e c t sum of f i e l d s i n 3 . Conversely, assume that every f i n i t e l y generated subring of R i s isomorphic to a f i n i t e d i r e c t sum of f i e l d s i n 3 . Then x = x f o r a l l x e R so R e Z . Thus, by - 1 5 5 -3 - 4 . 3 there are i d e a l s I : a e A of R such a n f l : a e A] = (0) and R/I i s a f i e l d of f i n i t e c h a r a c t e r i s t i c f o r a l l a e A . But then R/*a must be a f i n i t e f i e l d s ince x N - x = 0 e I f o r a l l x e R . There-f o r e , f o r each a e A , there i s an x Q e R such that (<x > + I )/I = R/I . But then R/I„ i s a homomorphic image of < x a> s o R / l a i s l s o m o r P n i c t o a f i e l d i n 3 . Q.E.D. 5 . 6 . 5 THEOREM: I f ft i s a c l a s s of r i n g s which i s not the c l a s s of a l l r i n g s then the f o l l o w i n g are e q u i v a l e n t : ( 1 ) ft i s a semi-simple r a d i c a l c l a s s . ( 2 ) There i s a f i n i t e set of primes T and f o r each p e T a f i n i t e C.U.D. set of p o s i t i v e i n t e g e r s S such that R e ft • i f and only i f R i s isomorphic to a sub d i r e c t sum of f i e l d s i n (P ^  : p e T and a e S •) . P y ( 3 ) There i s a f i n i t e set of primes T and f o r each p e T a f i n i t e C.U.D. set of p o s i t i v e i n t e g e r s S such that P R = ©C3p(S p) : p e T} . Proof: Let ft be a c l a s s of r i n g s which i s not the 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 that ( 2 ) and ( 3 ) are eq u i v a l e n t . - 156 -Assume that ft i s a semi-simple r a d i c a l c l a s s . Then by 5 . 6 . 2 and 5 - 6 . 3 & i s an elementary r a d i c a l c l a s s and ft <_ e ' . Therefore, by Theorem 3 - 4 . 1 4 , ft = © ( 3 p ( S p ) : P e T] f o r some set of primes T and C.U.D. sets of p o s i t i v e i n t e g e r s S p f o r each p e T . For each p e T , l - e S p . Let R = T R F p : P e T] . Since ft i s a semi-simple c l a s s R e ft <_ £' . Because R €•£' , R e FC so c e r t a i n l y T must be f i n i t e . Just as i n 5 - 6 . 3 we see that each S must be f i n i t e by c o n s i d e r i n g TTfF : a e S } . P p P Conversely, assume that ft = © ( 3 p ( S ) : p e T] where T i s f i n i t e and f o r each p e T , S Is a f i n i t e p C.U.D. set of p o s i t i v e i n t e g e r s . Then ft i s a r a d i c a l c l a s s . Since ft i s an elementary c l a s s ft s a t i s f i e s c o n d i t i o n (E). Suppose that every non-zero i d e a l of a r i n g R can be homomorphically mapped onto a non-zero r i n g i n ft . Then by 5 . 6 . 4 every i d e a l of R can be mapped onto a f i e l d i n { F : p e T , a e S } . From the proof of Theorem 46 i n P p D i v i n s k y [7] we see that t h i s i m p l i e s that R i s isomorphic to a su b d i r e c t sum of f i e l d s i n (F : p e T , a e S] . P Then, by 5 . 6 . 4 again, R e ft* = ft . Thus ft s a t i s f i e s condi-t i o n (F) . Therefore ft i s a semi-simple r a d i c a l c l a s s . 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 discussed 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. Amitsur, "The i d e n t i t i e s of P. I . - r i n g s " , Proc. Amer. Math. Soc. 4 , 2 7 - 3 4 ( 1 9 5 3 ) . [2] V. Andrunakievic, "Radicals i n a s s o c i a t i v e r i n g s I " , Mat. Sb., 4 4 , 179-212 ( 1 9 5 8 ) . [3] , "Radicals i n a s s o c i a t i v e r i n g s I I " , Mat. Sb., 5 5 , No. 3 ( 9 7 ) , 329-46 ( 1 9 6 1 ) . [4] V. Andrunakievic and Ju. M. Rjabuhin, "Rings without n i l p o t e n t elements, and completely simple i d e a l s " , S o v i e t Math. Dokl., V o l . 9 , No. 3 , 5 6 5 - 6 8 (1968) . [5] E. D. Armendariz, "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 ( 1 9 6 8 ) . [ 6 ] R. Baer, " R a d i c a l I d e a l s " , Am. J . Math., 6 5 , 5 3 7 - 6 8 ( 1 9 4 3 ) . [7] N. J . D i v i n s k y , Rings and R a d i c a l s , (Toronto: U n i v e r s i t y of Toronto Press, 1 9 6 5 ) . [8] L. Puchs, A b e l i a n Groups, (Pergamon Press, Inc., New York, i 9 6 0 ) . [9] D. Goldman, " H i l b e r t Rings and the H i l b e r t N u l l s t e l l e n -s a t z " , Math. Z e i t . 54, 136-140, ( 1 9 5 1 ) . [10] I . N. H e r s t e i n , "Theory of Rings", U n i v e r s i t y of Chicago Math. Lecture Notes, ( 1 9 6 1 ) . [11] N. Jacobson, S t r u c t u r e of Rings, (Am. Math. Soc. C o l l . - 159 -Publ. 3 7 , 1 9 6 4 ) . [12] I . Niven and H. S. Zuckerman, An I n t r o d u c t i o n to the  Theory of Numbers, (New York: John Wiley and Sons, Inc., I960). [13] Ju. M. Rjabuhin, " 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 of r i n g s " , Akad. Nauk. Moldav. SSR, K i s h i n e v , 1965-[14] G. T h i e r r i n , "Sur l e s ideaux complement premiers 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 Algebra, V o l . I ( P r i n c e t o n , N. J . : Van Nostrand, 1 9 5 8 ) . 

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