UBC Theses and Dissertations

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UBC Theses and Dissertations

Noetherian theory in modules over an arbitrary ring. Burgess, Walter Dean 1964

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NOETHERIAN THEORY IN MODULES OVER AN ARBITRARY RING by W a l t e r Dean  Burgess  B.A., The U n i v e r s i t y o f B r i t i s h  Columbia,  1959  A THESIS SUBMITTED IN P A R T I A L FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF Master of A r t s i n t h e Department of Mathemat i cs  We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o t h e required  standard  THE UNIVERSITY OF BRITISH COLUMBIA April,  1964  In presenting t h i s the  requirements  for  B r i t i s h Columbia, available mission  for  for  I  an advanced agree  reference  extensive  representatives.  cation  of  this  w i t h o u t my w r i t t e n  Department  of  by the  for  the  iJUwJ  ^  I  i s understood  further thesis  agree  that  that  freely per-  for. s c h o l a r l y or by  c o p y i n g or  f i n a n c i a l g a i n s h a l l not  permission*  /CH '  the U n i v e r s i t y o f •  Head o f my Department  ^ V ^ f i u ^ v ^ V i - c-C  z  at  f u l f i l m e n t of  L i b r a r y s h a l l make i t  this  The U n i v e r s i t y of B r i t i s h C o l u m b i a , V a n c o u v e r 8, Canada Date  in partial  degree  and s t u d y *  It  thesis  that  c o p y i n g of  p u r p o s e s may be g r a n t e d his  thesis  publi-  be a l l o w e d  ii  Abst r a c t Two  methods o f g e n e r a l i z i n g t h e c l a s s i c a l N o e t h e r i a n  to modules over first  arbitrary  i s by e x t e n d i n g  of Murdoch t o modules.  r i n g s are d e s c r i b e d i n d e t a i l .  the primary  f o r elementary  components  n o t i o n s o f r i n g and module  included f o r completeness.  sub-  The d e v e l o p m e n t i s s e l f - c o n t a i n e d  The d e f i n i t i o n o f p r i m a l subraodules  illustrations.  and i s o l a t e d  The  The s e c o n d i s by u s i n g t h e t e r t i a r y  m o d u l e s o f L e s i e u r and C r o i s o t . except  ideals  theory  Some c o n c r e t e  theory.  w i t h some r e s u l t s i s examples are g i v e n  as  Acknowledgement  I would l i k e topic  t o t h a n k D r . D. C. Murdoch  and f o r t a k i n g  supervise  my  work.  time d u r i n g  f o r suggesting  an e s p e c i a l l y  busy y e a r  to  the  iii  Table  of  Contents Page  Introduction  1  Notation  2  P a r t I . C l a s s i c a l N o e t h e r i a n T h e o r y and t h e T h e o r y o f Residuals  3  A.  C l a s s i c a l N o e t h e r i a n Theory  3  B.  Theory of R e s i d u a l s  4  Part  I I . G e n e r a l i z a t i o n s of the Decomposition  and t h e F i r s t  Theorem  U n i q u e n e s s Theorem  Ideals  12  A.  Primary  and P r i m a r y S u b m o d u l e s  12  B.  T e r t i a r y Submodules  18  C.  P r i m a l Submodules  28  D.  The R e l a t i o n s h i p Between t h e P r i m a r y , T e r t i a r y P r i m a l Submodules  P-art I I I . G e n e r a l i z a t i o n s o f t h e S e c o n d U n i q u e n e s s Theorem A.  I s o l a t e d Components  B.  A G e n e r a l Component f o r M o d u l e s w i t h  Part  IV.  References  E x a m p l e s and Remarks  and 31 32 32  (RCC)  35 42 48  Introduction. giving  Since  1950,  generalizations  t o noncommutative  of  lattices.  more i m p o r t a n t  of  of  the  and  rings  these  theory  and  submodules.  tive  Part  I contains  ideal  in  the  The  of of  are  theory ideal  parts two  the  if A  which  s u b m o d u l e s of  Lesieur  and  of  —  Murdoch are  brief  of  ideals  the  attentheory  Proofs  which  do  to m u l t i p l i c a -  parts. the  of  us  classical the  the  theory  ideals in a  to  Noetherian of  r e s i d u a l s of  the  residcommuta-  commutative  r e s i d u a l of A  avoid  and  of primary the  discussed  the  use  the  tertiary  i n Part  classical  submodules.  d e s c r i p t i o n of p r i m a l  ideals —  tertiary  d e c o m p o s i t i o n s ' * and  1  and  elements  words.  is called  allows  the  of  by  elements  thesis.  c i a t e d p r i m e s * theorems of both p r i m a r y  A}  the  restricted  generalized  B are  and  and  to m u l t i p l i c a -  developed  o u t l i n e of  major g e n e r a l i z a t i o n s  '•existence  be  o u t l i n e of  [c:c_B c  theory  Croisot  and  into four  i s an  rings  arithmetic  I have  c h a n g i n g a few  part  =  the  submodules.  e l e m e n t s may  theory  I have p r e s e n t e d  generalizes  generalizations  primary  of the  ideal  i n terms o f  thesis  i d e a l s and  (i.e.,  A:B  of  expressed  this  a brief  in this  It i s t h i s later  be  is divided  Residuals  ring, JB).  Also  published  theory.  arbitrary rings  m e r e l y by  thesis  uals. tive  use  This  theory.  may  latter  arithmetic  lattices  a unified  In most o f  the  Noetherian  thesis  approach  t o modules o v e r  involve  as  this  The  tion  not  In  i n terms o f  lattices.  the  classical  modules o r  tive  using  the  r e s u l t s have been  r i n g s , modules o v e r noncommutative  multiplicative  Noetherian  many new  submodules.  the  submodules  II.  The  of  extensions  ^uniqueness of  theory  The  the  part  are  given  asso-  for  concludes with  a  2  The e x t e n s i o n o f t h e " u n i q u e n e s s o f i s o l a t e d  components'*  theorem i s t h e t o p i c of P a r t I I I , The  final  p a r t c o n t a i n s some e x a m p l e s  remarks i n d i c a t e  and r e m a r k s .  The  some o f t h e p r o b l e m s w h i c h n e e d f u r t h e r  investi-  gation. Notation. used.  Throughout t h i s  thesis  U n d e r l i n e d L a t i n c a p i t a l s A, _B, ••" w i l l  ('•two-sided'  1  understood).  two-sided, l e f t s e t S.  The s y m b o l s  and r i g h t i d e a l s , r e s p e c t i v e l y ,  respectively.  S e t complements  Sum, u n i o n s and i n t e r s e c t i o n s [|jL:P],  ideals represent  g e n e r a t e d by t h e and p r o p e r s e t  are i n d i c a t e d  o f submodules  be  ( S ) , ( S | , |S) a r e t h e  The s y m b o l s c and c a r e , f o r s e t i n c l u s i o n  inclusion  {^LsP},  represent  P l a i n c a p i t a l s L, M, ... w i l l  o n e - s i d e d i d e a l s o r submodules.  modules  the f o l l o w i n g notation w i l l  by ^.  may be e x p r e s s e d by  [nL:P| w h i c h may be r e a d as " t h e sum o f a l l s u b -  L having property P ,e t c . tt  3  PART I A.  Classical  Classical  N o e t h e r i a n T h e o r y and t h e T h e o r y o f R e s i d u a l s  Noetherian Theory.  commutative r i n g s line w i l l An product  P of a commutative r i n g R i s c a l l e d prime  B c P.  is finitely  JL§ £ .Q and A The  of  i f the  An i d e a l Q i s c a l l e d p r i m a r y i f t h e p r o d u c t  R has t h e ascending  r(A)  out-  here.  ab £ .Q and a t Q i m p l i e s t h a t b  ideal  theory f o r  ab £ P and a t P i m p l i e s t h a t b £ P o r i f AB rz P and  A <£ P t h e n  If  ideal  i s so w e l l known t h a t o n l y a v e r y b r i e f  be g i v e n  ideal  Noetherian  Q implies  = { r £ R: r  £ Q f o r some p o s i t i v e  chain condition  generated  radical  n  f o r ideals  then  i n t e g e r n. every  so t h a t _Q i s p r i m a r y i f , and o n l y i f ,  Q Q f o r some p o s i t i v e  B  n  i n t e g e r n.  r ( A ) o f an i d e a l A i s d e f i n e d by n  £ A f o r some p o s i t i v e  integer n}.  The  radical  an i d e a l A i n a r i n g w i t h t h e a s c e n d i n g c h a i n c o n d i t i o n may be  expressed The  as r ( A ) = { ^ B :  Decomposition ascending  B  Theorem;  c A f o r some p o s i t i v e  n  In a commutative r i n g w i t h t h e  chain c o n d i t i o n , every  the i n t e r s e c t i o n If A =  integer n}.  ideal  c a n be e x p r e s s e d as  o f f i n i t e l y many p r i m a r y  fl •"" 0 Qy, i s an i n t e r s e c t i o n  ideals. as d e s c r i b e d i n  t h e t h e o r e m , s u p e r f l u o u s t e r m s may be d e l e t e d so t h a t t h e i n t e r section  i s irredundant.  radical  P, t h e i r  the i n t e r s e c t i o n t h a t the prime case  I f two p r i m a r y  intersection  ideals  have t h e same  i s primary with r a d i c a l  P.  Hence,  d e s c r i b e d i n t h e t h e o r e m may be a r r a n g e d so ideals  r  (^j)»  •••»  an i r r e d u n d a n t i n t e r s e c t i o n  r  C2 ) fc  a  r  e  distinct;  i s called short.  in this  4 The F i r s t  Uniqueness Theorem;  the  chain  ascending  D .2£, be s h o r t  condition.  ( % )  r  with  f| ••" 0 .Qj, = .QJ 0  Let A =  r e p r e s e n t a t i o n s o f an i d e a l A o f R as i n t e r -  sections of primary "'»  L e t R be a c o m m u t a t i v e r i n g  a  n  ideals.  Then k = k  r(2pf  d  and t h e r a d i c a l s  f  r ( g » ) c o i n c i d e i n some t  order. fl "•• fl 5  If A = ^ intersection a subset isolated  of primary  i s a short  t  i d e a l s and r C ^ ) = P  o f t h e p r i m e s P^, \  i fP  c P.  implies that P  i n t e r s e c t i o n Q, "l i s o l a t e d component o f A. The S e c o n d U n i q u e n e s s Theorem: ascending  short  chain  r 1 #  P.^ s a y , {P^ ,  The c o r r e s p o n d i n g  the  r e p r e s e n t a t i o n o f A as an (-Q )  =  k  P^ } i s c a l l e d  i s one o f t h e P. f) ••• Q  P. •  i s c a l l e d an ' j  L e t R be a c o m m u t a t i v e r i n g  condition.  P^;  L e t A <=  with  • •• n Qy. be a  r e p r e s e n t a t i o n o f an i d e a l A as an i n t e r s e c t i o n o f p r i -  mary i d e a l s .  Then t h e i s o l a t e d c o m p o n e n t s o f A a r e u n i q u e l y  determined; that i s , they  a r e t h e same f o r a l l s h o r t  representa-  t i o n s o f A. B.  Theory of R e s i d u a l s .  I n what f o l l o w s , some a s p e c t s  a r i t h m e t i c o f submodules w i l l a general  lattice  t i c e s of, L e s i e u r  setting  be d e v e l o p e d .  T h i s may be done i n  ( f o r example, the m u l t i p l i c a t i v e  lat-  and C r o i s o t [9])# b u t f o r t h e p u r p o s e s o f t h i s  t h e s i s modules are s u f f i c i e n t l y this  of the  general.  s e c t i o n may be u s e d i n t h e l a t t i c e  nificant modification.  A l l of the proofs of formulation without  sig-  5  Throughout  this  i d e a l s A, B, e . .  and M w i l l  ...  Definition  1;  section S w i l l  denote  be a l e f t  a general  R-module w i t h  left  well defined  (A + B)N C, L so t h a t  i f t h e s e t o f i d e a l s has a maximal  A j £ Ag c ••• be an a s c e n d i n g c h a i n i = h, 2, ••• .  and Z o r n ' s  called  L—*N i s  element.  Let  o f i d e a l s s u c h t h a t A^N c L  Then,  lemma g i v e s  D e f i n i t i o n 2:  the r e s u l t .  G i v e n a submodule L o f M and an i d e a l A o f R, t h e  s e t o f s u b m o d u l e s N s u c h t h a t AN c L h a s a m a x i m a l  element  c a l l e d t h e r i g h t r e s i d u a l o f L b_y A and i s d e n o t e d  L-t—A.  The  submodule L-s—A i s w e l l d e f i n e d  that  by an argument s i m i l a r t o  f o r L—*N. At t h i s  of  element  r e s i d u a l o f L by_ N and i s d e n o t e d b y L—»N.  I f AN c L a n d BN c L t h e n c l e a r l y  for  submodules  G i v e n two s u b m o d u l e s L and N o f M, t h e s e t o f  i d e a l s A o f R s u c h t h a t AN cz L h a s a m a x i m a l the  r i n g with  s t a g e i t i s u s e f u l t o n o t e some s i m p l e  r e s i d u a l s which w i l l  be u s e d f r e q u e n t l y . i n t h e s e q u e l .  (a)  A(L-s-A) c L, (L—»N)N CZ. L.  (b)  A  1  eg A  2  properties  implies  ( c ) - L, c L_ i m p l i e s  Lt—k^ c  L.-t—A  Lt—k^;  cz L„-s—A and L. —»N cz L — » N . 5  6  L-s-AB = ( L - ^ A ) - j - B .  (4)  BN a L-s-A and N c N  (Lf-A)-s-B  c  W(N  (e)  (L  • M )  1  2  + L )t-A  x  2  L*-(A For  I f N c L-s-AB t h e n ABN (L-s-A)-*-B.  L so t h a t  c  Conversely, i f  t h e n BN c L-s-A so ABN c L. = (L-fN ) n  (L-fN );  (L -A)"n  (L t-A);  1  2  1 +  2  2  • A ) = ( L < - A ) D (L+-A ).-'  1  2  1  2  many p u r p o s e s a module o v e r a g e n e r a l  r i n g i s t o o broad  a c o n c e p t and t h e f o l l o w i n g f i n i t e n e s s c o n d i t i o n  i s a useful  res-  triction: (RCC)  A left  condition  R-module M i s s a i d t o h a v e t h e r e s i d u a l  i f M has the a s c e n d i n g chain  r e s i d u a l s and R h a s t h e a s c e n d i n g c h a i n  condition  chain  on r i g h t  condition  on  left  to saying  that  residuals. By t h e u s u a l  a r g u m e n t , (RCC) i s e q u i v a l e n t  left  o r r i g h t r e s i d u a l s have m a x i m a l e l e m e n t s .  this  condition  will  I n what  sets of follows  be r e f e r r e d t o by a module M h a s ( R C C ) " o r n  a s i m i l a r phrase. Definition that  An i d e a l P o f R i s c a l l e d p r i m e i f AB c P i m p l i e s  3:  e i t h e r A c P o r B cz P.  An e q u i v a l e n t  formulation  aRb cz P i m p l i e s Definition of  hi  using  elements i s :  a £ P or b £ P ([10],  P i s prime i f  Theorem l ) .  An i d e a l A o f R i s c a l l e d a p r o p e r l e f t  a s u b m o d u l e L o f M i f A = L—*N f o r some N ^ L.  residual  7  Lemma 5 :  An i d e a l  residuals  P w h i c h i s maximal  o f a submodule  Proof:  left  Lemma 6 : proper  I f M h a s (RCC) then left  The n e x t ideal  if,  We  Hence,  which c o n t a i n s  3  but i s worth n o t i n g . e v e r y submodule has a maximal  lemma g i v e s a u s e f u l method left  L—? (L-s-A) = A where The c o n d i t i o n  residual.  Conversely,  L-t—(L—s-N) p N s i n c e  f o r d e c i d i n g i f an  residual.  An i d e a l A i s a p r o p e r  Proof:  B £ P.  residual.  i s a proper  Lemma 7 '  i s obvious  o f P.  residual  Thus, L—s-BN = L—s-N and so A cz P. The f o l l o w i n g  left  Assume AB c P w i t h  L by t h e m a x i m a l i t y  A c L—rBN and L—s-_BN i s a p r o p e r L—*N.  left  L i s prime.  L e t P = L-*N, N £ L.  have ABN c L and .BN  i n the set of proper  left  residual  o f L i f , and o n l y  L-s-A •=> L . clearly  implies  that  l e t A = L—s-N, H £  A i s a proper  L.  Then,  (L—s-N )N C L, but L-s-(L—sN) ZO L s i n c e N £ L.  We have L-*—A 3 L and L—s-(L-s—A) = A s i n c e  A c L-? (L-s-A) <- L—s-N = A. J | Theorem 8 :  I f M has (RCC) t h e n  corresponds P,P~ — 1—2  of  let  that  P M c L where P. i s a maximal p r o p e r —n — —1  left  residual  residual  o f L and  ... P, 1 .  —1—2 —  Proof:  L there  number o f p r i m e s P^ such  L-s-P,P  a finite  t o any p r o p e r , submodule  0  i  -  i  L e t P^ be a maximum p r o p e r  P, = L—s-N,.  Let  left  L-s—P. = K. and then K.  D  L by Lemma 7 and  8  P^  L — 2  a  left  L—*h,  residual  If  °* -2  Let Kg = K^f—Pg  + M we may  c o n s i d e r a maximum p r o p e r  a n d  =  K  l"~* 2 Y  L  2  * L+-P^Pg.  "*' 2. Y  2  L - r M  '  The p r o c e s s may  be c o n t i n u e d i n d u c -  t i v e l y and the p r o c e s s t e r m i n a t e s a f t e r f i n i t e l y the (RCC).  We have K  - M - Lt-P^  R  Pg ••• J?  Lemma 9:  I f M has  #  L and A tz\ L—t(Lt—A).  L t - ( L — » H ) 2 N so t h a t L-t—A  Proof:  proper l e f t  (L-J-B),  P C C.  (Lt-B)t-P s i n c e c l e a r l y P c C but C[ (L+-B)i-P], s (L-t-B)-t-P cz ( L t - B ) t - C . L—t(L-t—  C c P. The  BC)  cz  Lt-P  3  1  L s i n c e N ^ L.  r e s i d u a l of L and i f  r e s i d u a l of  Let C = ( L - t - B ) - • [ (L-t-B)-t-P] .  P[(L+-B)+-P] c  cz  z>  I f P i s a maximal p r o p e r l e f t  B gt P then P i s a  residual  C o n v e r s e l y , i f P i s a p r o p e r maximal  r e s i d u a l of L w i t h A c P = L—tN then L+-A  Lemma 10:  BC  r e s i d u a l of L,  I f L-t—A r» L then L—t(L-t-A) i s a p r o p e r l e f t  Proof:  •  result  (RCC), L-t-A p L i f , and o n l y i f A i s con-  t a i n e d i n . a maximal p r o p e r l e f t  left  and the  9  follows*  of  n  many s t e p s by  L+-B.  Since  C o n v e r s e l y , L+-BC = (L-t-B)-t-C -  (L-s-B)-t-C  s  ( L t - B ) t - P by the f a c t t h a t  (Lt-B) by d e f i n i t i o n of C so Then, (L+-B)t-P = L+-BP p L-t—P.  L—t(L-t-P) - P.  T h i s g i v e s the f i r s t  Thus  But, P i s prime and B ^ P so  c o n d i t i o n of Lemma  1.  r e s u l t f o l l o w s from Lemma 7 i f (L+-B)*-P 3 L-t-B.  Assume  ( L t - B ) t - P - L-t—J, then Lt-P Q L+-J. and B c L - t (L+-B) Q L—t (L-t-P) = P —  a contradiction.  J|  9  Theorem  11;  I f M h a s (RCC) and L ^ M t h e n L h a s o n l y  many m a x i m a l p r o p e r l e f t Proof: Theorem  such t h a t  residuals.  L e t P be a m a x i m a l p r o p e r l e f t  8 there i s a f i n i t e  r e s i d u a l o f L.  Z •  S i n c e P i s maximal i t i s  n  p r i m e and so f o r some i , P^ c P.  Assume P^ ^ P f o r  i - 1 . By Lemma 10, P i s a p r o p e r l e f t  r e s i d u a l of  = L-t—2.-^' S i m i l a r l y , P i s a p r o p e r l e f t r e s i d u a l o f L i — P . ••• P. . = K, S i n c e P. i s a m a x i m a l p r o p e r l e f t 1 i - i i - i — i of  K._  P  lf  Theorem the  P implies  c  11 i m p l i e s  maximal p r o p e r l e f t In  left the  l a t e r parts  residuals  P = P.. that  g i v e s such  residuals  8 include a l l  o f L.  o f t h i s t h e s i s we w i l l n e e d a c l a s s  which are prime i d e a l s  a class  residual  J|  t h e p r i m e s o f Theorem  c l a s s o f maximal p r o p e r l e f t  tion  By  c o l l e c t i o n o f p r i m e s P,, P„, •••• P — i —<i —n  P = L—*N 2 L—*M 2 P-^  j = 1,  finitely  and t h e c l a s s  residuals.  and i s v e r y u s e f u l  of proper  i s larger  The f o l l o w i n g  than  defini-  i n the characterization  of t e r t i a r y i d e a l s i n P a r t I I . Definition  12:  submodule K  £  L i f t h e r e i s a submodule N <£ L w i t h  P = L—»N and i f  N and K ^ L t h e n L-*K = L — t N .  Essential Theorem is  An i d e a l P i s c a l l e d an e s s e n t i a l r e s i d u a l o f a  residuals  13:  a r e , by d e f i n i t i o n , p r o p e r l e f t  Every maximal p r o p e r l e f t  an e s s e n t i a l r e s i d u a l o f L.  residuals.  r e s i d u a l o f a submodule  L  10  Proof: P  =  L—tK  If  K—*N  the  maximality  =  by  14:  L,  and  CN ^  L.  and  B c  Theorem  Let  be  =  Proof:  Let  L-s—JB^,  is  a  prime  element 3  D  a \ i j  Proof:  more i f ,  N  and  P.  P  L-*N  L  K c  residual  N then  of  L,  L - + K D  K—S-N  L,  L—s-N  but  J|  is  prime.  residual  pf  with  P.  is  C ^  P=>  Then,  essential,  and  If  many  BCN c  L  L-—s-CN = L—s-N =P  (RCC)  set  has  of a  Lemma  residual  of  N ^ .  let  L t - B ^  7«  N  =  2  process  residuals  M has  (RCC)  every  usefulness i f  the  it  is  M has  only  10,  proper element  every  Let  be  =  2  N ^ * - J  —  left  to  an  L  N j ,  a  Thus  i n f i n i t e  ^  of  maximal  M has  chain  (RCC).J|  contradicting  submodule  Let  element a  3  _B^.  only  residuals.  and  of  prime  leads  right  14  of  maximal  By  essential  of  o  L  residuals.  are  Theorem  apparent  submodule  left  i n f i n i t e  Lemma  left ]  1  an  by  every  proper  the  which  instead  of  BC c  C o n t i n u i n g the  16:  f i n i t e l y  The  By 3  submodules  Corollary  be  a  L .  N.  and  essential  (RCC)  prime  proper  of  L  left  residual  that  CN c  M has  many  =  an  proper  P.  f i n i t e l y  N^  P  assume  If  of  K £  essential  Since  15:  If  Every  and  L—*CN  residuals  low  maximal  L.  N ^  is  a  N £  Proof:  of  is  L—s-N w h e r e  Theorem  with  P  Theorem  theory noticed  submodules,  of  15  gives  the  residuals  that  a l l  two-sided  the  ideals  result.  as  JH  developed  proofs were  would used  here f o l -  11  throughout.  The  condition  (RCC)  t h e f o l l o w i n g e x a m p l e s when  (RCC)  i s seen  t o be u s e f u l  i f we  note  holds.  (1)  M and R s a t i s f y  the ascending chain  (2)  M and R s a t i s f y  the descending chain  (3)  M satisfies  both  (ACC)  and  (DCC)  (4)  R satisfies  both  (ACC)  and  (DCC).  condition condition  (ACC) (DCC)  12  PART I I  Generalizations First  A  Primary  ideals  Definition  1;  x £ R such  and p r i m a r y  A subset H of  that  Multiplicatively clearly  2:  elements taining  submodules  a ring  R is called  an m-system i f there  of  sets  radical  and  complements o f p r i m e  ideals  are  r ( A ) o f an  ideal  Ui  radical  a l l prime  An  A i s the set of  e v e r y m-system  r(A) of ideals  an  ideal  which  A i n R i s the  are m i n i m a l  element  a £ R i s said  prime  ( r p ) t o A i f i t c o n t a i n s an element  Lemma 6:  negation i s c a l l e d  An  not  i d e a l _Q i s s a i d  i f aRb c  and  x £ A.  right  t o be  An  prime  right  ideals.  the  right  prime  (rp)  ideal  B is  which  primary  primary  i s rp  (nrp).  a t r(Q) t o g e t h e r imply  I f R i s commutative,  usual primary  that  right  right  primary)  implies  t o be  A i f aRx c A  Def i n i t i o n 5 :  set of  ( [ 1 0 ] , Theorem 2 ) .  ideal  The  inter-  i n the  an  A.  con-  A.  containing A  Definition  the  exists  M.  r £ R w i t h the p r o p e r t y t h a t  The  section primes  The  r meets  Theorem 3:  to  the  Theorem  ( i i ) a, b £ M i m p l y t h a t  axb £  closed  Theorem and  ra-systems.  Definition  to  Uniqueness  ( i ) M = 0 or  either  of the Decomposition  (or  b £  ideals  just  Q.  are  13  Proof: primary  This follows  i f , and o n l y i f , e v e r y e l e m e n t n o t i n T(Q) i s r p t o J J ;  and from t h e f a c t ring  t h a t t h e r a d i c a l o f an i d e a l i n a c o m m u t a t i v e  i s the i n t e r s e c t i o n  taining  the i d e a l .  Theorem 7:  o f a l l t h e minimal prime  con-  the ascending o r the descending  then the r a d i c a l of a r i g h t primary  ideal  prime. T h i s t h e o r e m . Lemma 8 and Theorem 11 w i l l  the p r e s e n t form. will  be r e s t a t e d ,  this  thesis.  Lemma 8:  and p r o v e d ,  for ideals  d i v i s o r which  results later i n  the ascending or the descending  then every  i d e a l has a minimal  chain  prime  i s nrp to i t .  many r i g h t p r i m a r y  (i)  These  i n a more g e n e r a l s e n s e  I f an i d e a l A i s an i n t e r s e c t i o n  D e f i n i t i o n 9:  such  n o t be p r o v e d i n  i n [11].  P r o o f s may be f o u n d  I f R has e i t h e r  condition  of  finitely  ideals A -  5  r  n ••• n  Q  k  an e x p r e s s i o n f o r A i s c a l l e d a p r i m a r y r e p r e s e n t a t i o n .  A primary  r e p r e s e n t a t i o n ( l ) i s c a l l e d i r r e d u n d a n t i f no  i n t e r s e c t i o n .0^ D ••• fl Q^^i A in  ideals  H  I f R has e i t h e r  chain condition is  s i n c e D e f i n i t i o n 5 s t a t e s t h a t _g i s  i = 1,  k.  +i  The p r i m a r y  c a l l e d s h o r t i f none o f t h e i d e a l s intersection primary.  n  """  n  -S.  k  i  s  contained  representation ( l ) i s o b t a i n e d by t a k i n g t h e  o f two o r more o f t h e i d e a l s  Q^,  Qj, a r e  14  The  f o l l o w i n g t h e o r e m i s a summary o f  important  r e s u l t s which  ideals).  The  Theorem  10:  tion  R be  for ideals.  representation. tion.  g  =  k = t and  and  a r i n g with Let A  two  n •••  1  the  i d e a l w h i c h has  a short  primary  condi-  a primary representa-  representations  n £  {,}  - 9[  k  n •••  coincide  IV shows t h a t  sufficient  primary  ascending chain  n (r(Q^) : i=1,•••,k}  s e t s of prime i d e a l s  Example 1 of P a r t i s not  an  short  j = 1,  [r(Qj):  dition  be  the  (of  more  [11].  t o Murdoch  Then A has  I f A has A  then  f o l l o w from D e f i n i t i o n 5  t h e o r e m i s due Let  some o f t h e  i n some  the  order.  ascending chain  to guarantee the  existence  of  con-  primary  representations. The tive  ideas  lattices  be  described  in  the  of  [9].  i n the  be  a r i n g R,  Lemma 11 :  Proof: x £ P.  In what f o l l o w s t h e  r(A)  Let  =  the  i d e a l A—s-R i s an  multiplica-  generalization  l a n g u a g e o f m o d u l e s but  the  f o l l o w i n g remarks  useful for ideal  will  motivation.  containing  A.  r (A—s-R).  x £ r(A).  Then i f P i s a p r i m e c o n t a i n i n g  I f P i s a prime c o n t a i n i n g  r(A) c  r(A^-s-R).  Let  P be  r (A—s-R) t h e n A c  x £ A—s-R s u c h t h a t  P so  a minimal prime i n the  p r i m e s c o n t a i n i n g A s u c h t h a t A—*R ^ exist  extended to the  language of t w o - s i d e d i d e a l s are In  and  o u t l i n e d above can  x 4 P.  P.  That  set  i s , let  x £ A—*R i m p l i e s xR c  x £ P of  there A  A,  so  15  xRx cr A. is  Hence  Definition  12:  define  radical  If M i s a left  the r a d i c a l  of the i d e a l  L i s denoted  Theorem the  13:  I f M has  L—tM  I f L = M,  - «•,  A  ra  c k  c  r ( L  l  that  P . ro j  L—tM  I f L 1* , ,  n ••• n  Definition  0  L  n  follows  number o f p r i m e s  and P^ f  L—»M  2  k}  ...  there  P^,  JP er L — t M . fc  include  The  the minimal  =  (L -tM) 1  ••• n  r ( L  n  ) .  since  n  If M i s a left  satisfied.  cr  m  P^  = r(L.) n i  )  L )~»M  r(L)i s  I f L * M,  n  ( L  N  R-module,  — » M ) .  a  a submodule Q o f M i s  c a l l e d p r i m a r y i f a l l of the f o l l o w i n g e q u i v a l e n t are  of  Ln a r e submodules t h e n  n  This  15:  The r a d i c a l  ... JP^, cr L — t M c P^. If k and A c (~) P. . C o n v e r s e l y , i=l  since  t h e n A c P,  n •••  Proof: X  I, a f i n i t e  i n t h e s e t (_P^ : i = 1,  P  2.  m.  1  Lemma 14: a  ( L  k, s u c h  containing  L—s-M  P  i s the  r(L—tM)  A such t h a t A  M—tM = R and r ( L ) = R.  i  primes  as i n D e f i n i t i o n  of the s e t of i d e a l s  by Theorem 8, P a r t  primes  that  and L i s a submodule o f  (RCC) and L i s a submodule o f M,  exists  minimal  R-module  o f L as r ( L — t M ) where  f o r some i n t e g e r  Proof:  = 1,  c o n t r a d i c t s the f a c t  by r ( L ) .  maximal e l e m e n t  L—tM  P which  9  prime.  M,  xRx c P, x t  conditions  16  A L c Q, L £ Q i m p l i e s A cz r ( Q ) .  (i) (ii) In  A^  r ( Q ) i m p l i e s Q-s-A = Q. 13, t h i s  v i e w o f Theorem  d e f i n i t i o n may be r e w o r d e d i n t h e  c a s e where M h a s (RCC) t o (iii)  A L c Q, L  Q then there e x i s t s  such t h a t A M  if,  16:  £  I f M h a s (RCC) t h e n Q M  Q h a s e x a c t l y one p r o p e r p r i m e  prime  residual  Q-s-M.  Proof:  i s p r i m a r y i f , and o n l y left  residual  i s minimal i n the s e t of ideals  T h i s prime  By Lemma 9 o f P a r t  Hence P = r ( Q ) and so _P i s u n i q u e  left  a unique minimal prime o n l y proper prime  Lemma 9, P a r t  I , Q+-P 3 Q so P c r ( Q ) .  and s i n c e e v e r y p r i m e  left  residual, i f A  A p r i m a r y submodule w i t h  P-primary  of f i n i t e l y  i f P i s prime.  (Q  1  r(Q  x  0 0-2^*""-  Q  E  =  Q  l  0  Q  2  s  o  Q  1  P t h e n Q+-A = Q by  J|  radical P will many P - p r i m a r y  l  0  and Q  1  Q  2  i  s  Q Q ) = r ( Q ) 0 r ( Q ) = P. 2  If Pi s  be c a l l e d  P-primary.  submodules i s a g a i n  T h i s i s t r u e s i n c e i f A £ P,  Q^-s—A = Q , Qg+^-A = Q i f 1  con-  Conversely, i f P i s  c o n t a i n i n g Q—s-M t h e n P = r ( Q ) .  I . So, Q i s primary.  The i n t e r s e c t i o n  containing  r e s i d u a l of a primary  t a i n i n g Q—s-M c o n t a i n s r ( Q ) , P i s m i n i m a l .  the  and t h i s  i s r(Q).  L e t P_ be a p r o p e r p r i m e  submodule Q ^ M.  integer k  Q.  k  Theorem  a positive  2  2  are P-primary.  P-P  r i r a a r v  since  Then,  17  Definition tion  17:  of  If  a  f i n i t e l y  submodule many  primary  _L  Q  (2) such The  an  expression  expression  section  Q  in  The  Q^.  the or  Lemma  n  1  (2) •••  of  the  18:  If  M has  i  =  1,  Proof c  L cr  Q  AN  c  Q.,  N £  i 0  (ii) Theorem A M m  A  m  (Q  cr  Q  2  0  13, x  .  (if  m >  l ) .  A ~ (Q  2  Lemma  1 9 I n  ra  1  L, L  so  A c  r(Q.)  there  is  an  w i l l  and  k  c  L-s—A  But  the  by  primary.  an  L  i f ,  contained  if  none  of of  two  primary.  and  primary  only  i f ,  A gt  such  that  P^  r(Q^).  exists  f)  N g£  ••• D Q P.  i ,  since  Q.  assume m  L.  is  of  or  A " (Q  minimality  of  m  1  2  Q..  for  Then,  minimal)  Q  m  3  i .  by  such  that  A M ro r a  either  n  implies  some  primary.  P^.  Now,  1)  m  L ±  A c  (assumed £  L  so  k  m =  fl Q ) fc  cr  L-t—A  that  H  with not  is  k  inter-  irredundant  minimality  (if  no  intersection  the  module  submodules  =  L-s—A  that  Q  short  the  is  =  =  some  k  a  (2)  1  -if  Qj, i s  L-t—A  Q  intersec-  representation.  n  called  Q^,  P^  =  •••  taking  integer  show  Q)  k  is  there  N £  n ••• n Q ) £ L .  primary not  where  for  Q n ••• n Q  PI  and  then  P,  n  2  L  A c  We  primary  D  But, Q.  If  irredundant  L-t—A cj:  If  AH  called  by  an  k  is  (RCC)  k  Q  a  (2)  as  submodules  called  H  of  expressible  is  submodules  representation  is  n ••• n  1  obtained  more  for  =  expression  submodules  L  (RCC),  a l l  of  an  which  irredundant have  the  intersection  same  radical  of is  18  Proof:  It i s sufficient  s u b m o d u l e s Q^, Q r(Q )  =- P ,  x  r  1  ^  2  which  2  ^  =  P  t h e n P c P j (1 P » 2  2'  Q  a r e n o t r e l a t e d by i n c l u s i o n . L e t =  l  Q  n  Q  and A ±  1  2*  A  s  V. '  m  e  Q  2  i s now p o s s i b l e  1  = P « 2  I f M i s a module w i t h  => Q.  r(Q) since Q J| theorem  so i t w i l l  I I I and  j u s t be  (RCC) and i f L h a s a p r i m a r y  t h e n any two s h o r t  L h a v e t h e same l e n g t h  primary representations  T e r t i a r y submodules.  The d e f i n i t i o n  s u b m o d u l e s i s due t o L e s i e u r these ideas  and C r o i s o t  i n some  of t e r t i a r y  order.  i d e a l s and  and t h e y h a v e  developed  i n t e r m s o f e l e m e n t s and i n t h e l a n g u a g e o f m u l t i p l i -  lattices  ( [ 5 ] , [ 6 ] , [ 7 ] and [ 9 ] ) .  below i s i n terms of t h e l a t t i c e modules o v e r a r b i t r a r y r i n g s . already  of  and t h e r a d i c a l s o f t h e p r i m a r y com-  p o n e n t s o f t h e two i n t e r s e c t i o n s c o i n c i d e  is  2  here.  representation,  cative  8  b u t i t i s a c o n s e q u e n c e o f Theorem 21 o f P a r t  Theorem 2 0 :  B.  r e s i d u a l of Q  t o prove the f i r s t uniqueness  a s p e c i a l c a s e o f Theorem 30 o f t h i s p a r t stated  P-primary,  s  Q so P j c P  S i m i l a r l y P_ cz P and P = P  is primary.  i  Q-J-j^ 3 Q and Q-i-.E  Therefore,  2  1  directly,  u  I f Q-s—A = Q t h e n Lemma 18 i m p l i e s  H e n c e , f o r some L^, P ^ L cz Q, ^  It  s  The i d e a l P i s a p r o p e r l e f t  by Theorem 16, so Q-t-P => Q. that A £ ?  t o show t h e r e s u l t f o r two p r i m a r y  Most o f what i s q u o t e d  formulation  The d i s c u s s i o n  a v a i l a b l e i n E n g l i s h [4]«  specialized to left i n terms o f elements  19  Definition  21:  A submodule Q o f an R-raodule M i s s a i d t o be  i f Q-e—A 3 Q and (Q-s-A) n L = Q i m p l y L = Q.  tertiary  E v e r y n - i r r e d u c i b l e module i s c l e a r l y and X d e n o t e  A submodule Q i s t e r t i a r y i f  Q i — A £ 0 i m p l i e s t h a t Q-e—A i s l a r g e  Definition  Let M = M - Q  t h e image o f a s u b m o d u l e X o f M u n d e r t h e n a t u r a l  homomorphism o f M i n t o M.  section with  tertiary.  every non-zero  22:  The t e r t i a r y  ( i . e . , has non-zero  inter-  submodule o f M ) . r a d i c a l o f a submodule L o f M,  d e n o t e d by ?v.(L), i s t h e maximum i d e a l o f R o f t h e f o r m A where (L-s—A) 0 N c L i m p l i e s N c L. In P a r t the  IV an e q u i v a l e n t  definition  e l e m e n t s o f t h e module and r i n g .  o f X(h)  the  does t h i s  Theorem 2 3 : the  i t is  and so we w i l l n o t  D e f i n i t i o n 22 i s m e a n i n g f u l by u s i n g  a r i t h m e t i c o f s u b m o d u l e s and i d e a l s .  theorem  i n terms o f  With t h a t d e f i n i t i o n  c l e a r t h a t A.(L) i s a w e l l - d e f i n e d i d e a l , a t t e m p t t o show t h a t  i s given  only  However, t h e n e x t  I n t h e c a s e t h a t M h a s (RCC).  I f M h a s (RCC), L a s u b m o d u l e o f M, t h e n K(L)  is  i n t e r s e c t i o n o f t h e e s s e n t i a l r e s i d u a l s o f L.  Proof:  I t f o l l o w s i m m e d i a t e l y f r o m D e f i n i t i o n 22 t h a t  i s t h e maximum i d e a l o f t h e f o r m A where  \(L)  (L-s—A) n N = L i m p l i e s  N = L. (i)  L e t P be an e s s e n t i a l r e s i d u a l o f L.  There  i s N z> L  s u c h t h a t P = L—?N and L c K c N i m p l i e s t h a t L-*K = L—»N. A c A.(L) and s a t i s f y  the condition stated  above.  Let  20  A g L - t ( L - t - A ) c L - t [ ( L t - A ) 0 H]  (3) L c  so  N  (L-s—A)  3  fl H  L c (L-t—A) n N c N  so  this  with  P  P  L  of  (3)#  and  so  A c K(L)  L  by  the  condition  on  A.  L - * [ ( L t - A ) f| N] = L — t N and  is  so  ez P  X.(L)  contained  in  for  Hence,  =  P.  every  their  Combining-  essential  residual  intersection.  k  (ii)  Let  P  JL =  where  i  P  P  1 #  are  f c  a l l  the  essential  i = l residuals a  L.  of  proper  left  residual  L—tN  is  then  P  =  Lt-X  2  L-s—P 3  that  N => L.  possible we  have  N  .  r  r+1  the  If  to  N N  r  such  ascending  is  an  of  L.  N  (L-t-l)  chain  cz N, N*  1  Again  I_ c  0  L  N  =  3'  P,  residual  ( L t — I)  =  does  not  N^ c N  such  that  .  r+1  of  L  L  3  right  such  N^  L  r>  not  residuals P  =  L—tK = L—tN  L—tN  an  also  Thus  r  the  then 3  .  By ' so  _I  -  is  Then, fact  it  is  L—tN.  If choose  the  (RCC)  that  there  essential N»  =  A  L—tN  L—tK = L—tN^  L—tN  3  is  1  P.  L—tN^  and  imply  and  L.  a n d J_ e  stabilizes  L—tN N*  3  N*  implies  L-^K =  r+1  L-t-P 3  contradicting  Then,  contradicting  L  L-*N ^.  and  that  L t - l 3  =  imply  L cz K cz N^ d o e s  N  L c N.  L  of  Cl N  K c N  that that  N  and  L ez K ez N  If  essential  L c  choose  such  r  an  L.  of  whence  N  (L-t—J.) n N = L  Assume  residual  (L-t-l)  satisfies  f) N* ez the  k  condition  and  A.(L)  =f  P. •  \ i =l  It of and  is  tertiary the  now  possible  submodules  f i r s t  J|  1  to  develop  which  uniqueness  w i l l  theorem  the lead  with  more  important  to  decomposition  a  only  (RCC)  as  tion. Lemma  2k;  If  L^,  L^  are  submodules  of  n •»• n L ) 3 K C L ^ n • • • n * . ( L ) . k  k  M  then  an  properties theorem  assump-  21 j Proof: exists  N such  n  (L -s-A) 1  L  l  L  n  Thus  2  -  l '  it  Q-s—A  =  -  L  n  L «  Hence, This  2  n  Assume  Lg.  N  £  =  gives  l  L  A c L  0  2  c  H  2  since N  there  Now  (L +-A) n  fl  i * ~ ^  A.  X ( L S  L  2  ) «  1  ;  a  n  d  L  n N c  (L -s—A) 2  A cz \(L^  fl  L  2  1  ) .  J|  does  not  hold  even  with  (RCC)  ([5],  ~k(Q)  and  Q is  submodule  (ii) i f ,  A c  If  Q has  (i) by 22  then  Q of  M is  or,  \{Q)  M has  (RCC)  exactly  one  tertiary  i f ,  equivalently, a  submodule  essential  and  A £  Q is  only  i f ,  \(Q)  implies  tertiary  i f ,  residual  P  and  If  Q is  tertiary  Definition gives  A c  21,  (Q-s-A)  k(Q).  Q-s—A zo Q a n d  and  A N cz Q,  n  K =  Conversely,  (Q-s-A)  n  L  =  N £  Q  then  Q implies i f  K =  A N cz Q ,  Q imply  A c  Q  N £  X.(Q)  and imply  Q so  L  =  Q  t e r t i a r y .  (ii)  If  Q so  unique  essential  A  P.  Q so  A c  Q is  P c  and  tertiary  A.(Q).  equal  residual  residuals,  Q is  L^  lemma.  Q imply  Q.  Definition  so  l -  L  n  1  (l^i-A)  h^r-k.  cz  2  N c  the  N £  Proof:  left  that  (l)  only  A.(Q)  3  l>  N cz L  fl (L2*~-^  (Lj^-s-A) N Q  —  N  =  2  so  2  ^  fl X . ( L )  n  2  L  )  1  L )-s-A]  ^ 2"*~-^  O  \ ( L  n L )i-A  1  inequality  25:  cz Q ,  Q-s-A 3  S  (L  n  let  7.1).  Theorem  and  =  n N =  gives  reverse  example  is  d  follows  Induction  Qt-P  n  and  2  [ (L^^  implies  2  AN  that  a  2  and  =  2  (L -«-A) ?v.(L )  k  (L t-A) L  A c  The  Let  tertiary  P  to of  But,  Q,  (i).  be  P  is  Theorem  ?v,(Q).  must by  and  A.(Q)  an 23  essential gives  Conversely,  if  =  residual P  \(Q)  a  P  the  is  so  of  that  Since  Q has  maximal  proper  maximal.  Hence  A £  implies  Q-s—A =  S  P  unique  P.  P  Q,  Q  22  As  an i m m e d i a t e  tertiary tiary  corollary  t o t h i s theorem,  ( R C C ) implies that the  r a d i c a l o f a t e r t i a r y submodule i s p r i m e .  and X ( Q ) = P we w i l l  Lemma 2 6 :  If  ^1 ^ ^ 2 *  and Q  2  call  If Q i s ter-  Q P-tertiarv.  a r e two P - t e r t i a r y  S i m i l a r l y f o r any f i n i t e  s u b m o d u l e s , so i s  number o f P - t e r t i a r y s u b -  modules. Proof: t h e n N <^ Q  X ( Q  1  n Q  i m p l i e s A cr P. A =  X . ( Q  1  n  Definition  Q  2  n  X  2  Q  =  X . ( Q  X  )  fl  ?v.(Q ) 2  = P.  I f AN cr 0^ n  or N £ Q  1  2  »  Either  Q  2  case  This i s true, i n p a r t i c u l a r , f o r I  ) .  27:  I f a s u b m o d u l e L c a n be e x p r e s s e d as an  L =  Q  1  n ••• n  Q  K  (4) i s c a l l e d a t e r t i a r y r e p r e s e n t a t i o n  intersection  intersec-  many t e r t i a r y s u b m o d u l e s  (4)  are  )  implies either N £ Q  tion of f i n i t e l y  then  2  i s i r r e d u n d a n t and t h e i d e a l s  o f L.  A . ( Q ^ ) ,  d i s t i n c t then t h e t e r t i a r y r e p r e s e n t a t i o n  I f the X ( Q ^ )  (4) i s c a l l e d  reduced. The d i s t i n c t i o n between t h e d e f i n i t i o n o f " r e d u c e d * and t h e d e f i 1  nition  of "short * i n D e f i n i t i o n 1  a l l o w us t o n o t e t h a t representations  the f i r s t  i s a special  17 i s i m p o r t a n t s i n c e  i t will  d e c o m p o s i t i o n theorem  f o r primary  case o f t h a t  for tertiary  decomposi-  tions. Theorem 2 8 : tertiary  I f M has ( H C C ) then e v e r y submodule has a r e d u c e d representation.  23  If  the ascending  chain  c o n d i t i o n f o r s u b m o d u l e s i s assumed t h e n f r o m Lemma 26 s i n c e e v e r y  the t h e o r e m f o l l o w s i m m e d i a t e l y cible the  submodule i s t e r t i a r y .  result  F o r t h e more g e n e r a l  f o l l o w s f r o m Lemma 37 b e l o w .  section C of t h i s  reduced  part.  tertiary  with  and Q  If  2  t  h  e  n  L  =  l  Q  n  2  N  H  (Qit-Pi) 0  expressions  2' "-1 l  =  Q  This for  2"  T  since Q  and, L  N  X  n  N  X  Q  =  h  0  s  g  v  2  n  N  =  2  N  e  X  P  m  s  l  s  ^l"  i  e  2  1  n N . 2  n o t , P^ i s c o n t a i n e d i n  n c e  2  respectively, i f  2  L = N  s  w h i c h must be P » 2  Intersect the last  1  n o t i n g t h a t N^-s—P^ p N^ and  2  8  "-^  -  n  N  fl ^  l  N  n  £  2  =  Q  2  s  0  N  l  0  N  2  Theorem 3 0 : tertiary  =  L  -  Q  l  o  3  n N . 2  lemma i s t h e t o o l w i t h w h i c h t h e f i r s t u n i q u e n e s s reduced t e r t i a r y  Hence,  2  Q fl ( N - s - P ) .  =  i sP^-tertiary,  1  i  ®2 *  =  ( N J L - S - P J ^ )  i  N  r e s i d u a l of Q  w i t h N^ fl N i  s u b m o d u l e s , o f a module M  r a d i c a l s P^ and P  2  8 -  a maximal p r o p e r l e f t Lf-Pj-,-  l  We have Qg" " 2-i  Proof:  two  implies the existence of a  are t e r t i a r y  (RCC), w i t h t e r t i a r y  £ L  x  representation  (RCC), t h e e x i s t e n c e  representation.  Lemma 29:  p  o f Lemma 37  and d i s c u s s e d i n  N o t e t h a t s i n c e Lemma 26 d i d n o t r e q u i r e of a f i n i t e  c a s e o f (RCC)  The p r o o f  i n v o l v e s p r i m a l submodules which a r e d e f i n e d  irredu-  theorem  r e p r e s e n t a t i o n s may be p r o v e d .  I f M h a s (RCC) and a submodule L h a s two r e d u c e d representation L  =  Q  1  D  n  Q  K  =Qjn  ••• n Q  m  24  t h e n k = m and t h e s e t s and  ]P« : P» = r ( Q J )  Proof:  (ii)  {P^: P^ =  1 < j < ra} c o i n c i d e  f  It i s sufficient  1 < j < nit (i)  of primes  t o show t h a t  i n some  x  £ P» o r P» £  P £ P» o r  P  < i < k}  order.  P^ = P_j f o r some j ,  Assume P^ i s d i f f e r e n t f r o m a l l t h e P*.  P  1  r ( Q ^ ) ,  Then e i t h e r  x  V ±l  x  2  x  . . . (•) By L  either =  L =  S  ± IL o r P  x  m  £  P..  a l t e r n a t i v e o f ( i ) and Lemma 29,  Q*  n ••• n  Q »  n • • • fl  Q  n  Q  2  n  Q  2  F F L  n ••• n  Q  n ••• n  Q  K  But  .  Q  =  K  L  fl • • • n  Q  and so e i t h e r  K  a l t e r n a t i v e o f ( i i ) p e r m i t s t h e use o f Lemma 29 t o g i v e fl • • •  L =  L = Q £ ft  •"•  n Q  M  0 Q  2  0 ••• PI Q -  fl Q  M  0 Q  2  n  (r) gives L = Q * L = Q  2  fl  tertiary  D Q  +  1  K  with  K  Let L =  N.  K  »  By  induction,  the irredundancy of a reduced  associated  be c h a r a c t e r i z e d  in  residuals. n • • • fi Q  F C  be a r e d u c e d t e r t i a r y  ( R C C ) .  representa-  L e t P = L—»N be an e s s e n t i a l  L c K c N implies  Then, t h e r e e x i s t s e x a c t l y  n  0 ••" D Q  decomposition w i l l  r e s i d u a l of L such t h a t  Q.  2  the uniquely determined primes  t i o n o f L i n a module w i t h  L =  D Q  a l t e r n a t i v e of  J|  representation.  terms of e s s e n t i a l  M  contradicting  a reduced t e r t i a r y  Lemma 31;  fl Q , r < m, e i t h e r  H ••• D Q  In what f o l l o w s ,  Given  K  one t e r t i a r y  L—*K = L—s-N = P.  submodule  such  that  25  Proofs Q± Q. X  PI  h Q  D  l  We h a v e H D L so ^ D H = L where  m  HQ.  N» = Q  0  fl ••• fl Q  M  m  fl N 3 L.  L—*N * - 0^—sN» s i n c e AN  unique e s s e n t i a l P K 1  [Q^,  Q}  Q  1  i s minimal  m  But ?  residual  1  = KiQ^  1  tertiary Q  2  n  Theorem 3 2 ; tation -  L .  Let L =  fl  Proof:  1  • • • fl Q  k  3  P  and P - J ^ .  x  If  r  - '  be a r e d u c e d t e r t i a r y r e p r e s e n -  (RCC).  By Lemma 3 1 ,  Then, t h e s e t o f p r i m e s the set of e s s e n t i a l  = Q  every e s s e n t i a l  To show t h a t 2  residual  P^ i s an e s s e n t i a l  D " °» H Qj, and c o n s i d e r a l l  m a x i m a l e l e m e n t P_ = L—s-K, L cr K cz L*. an e s s e n t i a l  = K 3 L.  i s P^-  J|  L o f t h e f o r m L—s-N where L cz N c L».  tion,  since  o f L.  set [ P ^ l i  consider L  and K c N* t h e n  fl N = L and  1 < i < k} i s e x a c t l y  X ( Q . ) :  1#  the d e f i n i t i o n of a reduced  Hence  o f L i n a module w i t h  residuals  the  n N 3  K  Q  (changing o r d e r o f terms i n t h e i n t e r s e c -  contradicting  representation.  ••••• n Q  |P.  2  c  i sthe  x  1  1  Now,  Therefore,  P = L - t N = L - t C d - * - ^ ) n N*] 2 L-S-^L-J-PJ^) m > 1 t h e same argument  3 P since' Z  But i f P^K cz  o f Q^.  H e n c e , L cz (L«—P^) fl . N * cz N.  tion) gives P = P  (i.e., i f  3 L g i v e s L—s-N * cz 0^—frN» b u t i f A N  l <  tertiary.  Let  S i n c e N 3 L, m > 1 and  L so ( I r t - P j ) n N» = ( Q _ P ) Q N» £ Q  £  ft N = L.  fc  Thus L e H » e N and P = L-sN». • . ——  c L by d e f i n i t i o n o f N».  1  fl Q  ft N = L t h e n r > m).  r  a  fl  residual  B u t , by Lemma 3 1 ,  o f L.  i s included i n  residual left  o f L,  r e s i d u a l s of  By ( R C C ) , t h i s s e t h a s a T h i s i s , by i t s c o n s t r u c -  K fl Q  2  fl ••• Ct Q  fc  = K fl L*  fl K = L f o r some i , i must be  T h e r e f o r e L = Q. fl K and £ = P,.  By r e n u m b e r i n g , t h i s method  1.  26  shows t h a t each of t h e primes P^ r e s i d u a l of L,  Pj, i s an e s s e n t i a l  r  J|  T h i s very u s e f u l theorem has many c o r o l l a r i e s , two o f which are noted here* C o r o l l a r y 33: - Z  n ••• n P .  x  k  Proof: of L.  Under t h e hypotheses o f Theorem 32,  X.(L) i s t h e i n t e r s e c t i o n o f t h e e s s e n t i a l  (Theorem 2 3 ) .  Corollary include  34:  j|  Under t h e hypotheses o f Theorem 32, P^#  Pj,  the maximal p r o p e r l e f t r e s i d u a l s Of L,  I t now can be shown t h a t for  residuals  any submodule L.  i n a module w i t h (RCC), r ( L ) g \ ( L )  By d e f i n i t i o n , r ( L ) i s t h e i n t e r s e c t i o n o f  the m i n i m a l primes c o n t a i n i n g  L—tM.  Except i n t h e t r i v i a l  case  where L = M t h e s e primes are p r o p e r i d e a l s and Theorem 8 o f P a r t I and Theorem 13 show t h a t The r e s u l t f o l l o w s  these primes are p r o p e r l e f t  i f i t can be shown t h a t every e s s e n t i a l  d u a l c o n t a i n s one of t h e m i n i m a l p r i m e s . of the (RCC).  residuals.  Let £  55 1  L—tt^  c h a i n of p r o p e r prime l e f t  r> P  T h i s i s a consequence  » "~* 2 L  2  N  3  ""  b  e  a  d  e  s  c  e  n  d  C l e a r l y P^ «* L—t(L-t—P^) so t h a t  L+-P  (  c L*~P  c  a  n  d  t  h  e  2  i  n  3  r e s i d u a l s of L which are c o n t a i n e d i n  an e s s e n t i a l r e s i d u a l P. 1  resi-  R C C  ) implies  t h a t the c h a i n  stabi-  l i z e s and the s e t of primes has a m i n i m a l element. In a module M w i t h (RCC) a p r i m a r y submodule has e x a c t l y one prime p r o p e r l e f t  residual  (which i s minimum i n t h e ' s e t " o f ' l e f t  27  residuals  containing  essential  since  L—tM).  This  e v e r y s u b m o d u l e has  r e s i d u a l s w h i c h are p r i m e .  Hence a p r i m a r y submodule i s t e r -  L =  fi  r(Q^)  f r(Qj) i f i ^ j i t i s a reduced t e r t i a r y  Since  the primes r(Q^)  30,  we  radicals coincide  in this  = X(Q^)  conclude that  are the  uniquely  The  and title  a l s o i s not of t h i s p a r t  a g e n e r a l i z a t i o n of  the  Theo-  i s not  two  primary  implies that  tertiary  submodules  (RCC)  i s shown  R-module w i t h  (RCC)  That  are this  by  and  i f R is  i f , and  only i f ,  i s primary. Proof:  sufficient  Lesieur condition  and  I t i s as  if  implies there  N c AL  a left  Croisot  follows:  R-module M has  ([9],  p.  74)  have g i v e n  a  t h a t e v e r y submodule which i s t e r t i a r y  primary.  as  representation.  c l a s s i c a l p r i m a r y submodules.  If M i s a l e f t  since  i n t e r s e c t i o n of  distinct  commutative t h e n a submodule L i s t e r t i a r y it  then  tertiary.  i s t r u e , at l e a s t f o r m o d u l e s w i t h Theorem 33:  If  d e t e r m i n e d by  irredundant  p r i m a r y s u b m o d u l e s whose r a d i c a l s a r e (Lemma 1°)  case.  fl Qj, i s a s h o r t p r i m a r y r e p r e s e n t a t i o n  may  be  many e s s e n t i a l  and  •••  two  finitely  tiary  rem  the  u n i q u e p r i m e r e s i d u a l must  (i)  an  e x i s t s L» c (RCC)  a sura o f M - p r i n c i p a l  and  ideal A i s called M-principal L s u c h t h a t N = AL»,  e v e r y i d e a l o f R can  i d e a l s , then every t e r t i a r y  be  ( i i )i f expressed  submodule i s  primary. We  show t h a t  principal Clearly  every commutative r i n g R i s such t h a t  i d e a l s are  M-principal  every i d e a l i s the  sum  is  f o r e v e r y module M o v e r  of p r i n c i p a l i d e a l s .  the R.  28  (a)L 3  N,  a e R.  (a)L* c  N.  But,  (a)m c N i f , and  e N,  conversely  Let Clearly (a)m c N and  i m p l i e s am  (ia)m = i(am)  (a)m cz N. But  n e  + N,  L»  = N-s-(a) =  am  [m e M:  only  i n t e g e r i ; the  —  and  C.  Primal  [5]  and  ideal  known i f e x a m p l e s e x i s t  i s not  The  i d e a of  above  gives (a)L'.  requires  of modules  a primal  a g e n e r a l i z a t i o n of the  i d e a l s which are  However, e v e r y  s e c t i o n the  N  (without  submodules  primary.  thearc a r e p r i m a l  Example 3)»  o  3  s i n c e even i n commutative r i n g s w i t h  condition  (Ra)m  N}.  implies  c o n d i t i o n r e f e r r e d to i n the proof  submodules.  [9])  N:  commutively  (RCC)) o v e r c o m m u t a t i v e r i n g s w h i c h h a v e t e r t i a r y w h i c h a r e not  e  i s n e M such t h a t n t  there  a contradiction.  i t i s not  (a)m c  i f , am  e M implies  ( a ) L so n = ( r a * i a ) r a , m e L, w h i c h  sufficient  (RCC)  e N f o r any  Thus i f (a)L»  rra • im £ L* The  Let  definition  primary  of p r i m a l  e l e m e n t a r y r e s u l t s because of t o p i c s of t h i s  thesis.  classical  primary  the  ascending  not  primary  ideal, i s primal.  chain (Part  In  close connection  r e s u l t s are q u o t e d  with  34s  A submodule L o f  a left  without  R-module M i s  ©  primal (L-s-A^  i f L - s - z > L and n  (L-s-A.) 3  Clearly, f)-irreducible  L - J  —A  2  D L imply  that  L. submodules are  primal.  some  other  proofs. Definition  IV,  this  submodules i s i n c l u d e d w i t h  their  Several  ([1],  submodule  called  29  Theorem 35s  I n a module w i t h ( R C C ) ,  a submodule L i s p r i m a l i f ,  and o n l y i f , i t h a s e x a c t l y one maximum p r o p e r l e f t Proof:  I f P i s the unique  maximum p r o p e r l e f t  residual of  L t h e n L-s—A^ 3 L i m p l i e s A cr P by Lemma 9 o f P a r t I , L, L-s-A 3 L i m p l y A  L-s—a  cr J P , • A ' cr P  ±  0  2  Thus, (L-s-i^) n (L-8-A ) = L * - ( A | * A 2  g  2  ) 3 L.  and P_ a r e two d i s t i n c t m a x i m a l p r o p e r l e f t 2  L-s—Pj, 3 L and L-s-P_ => L b u t (lU-P^ + P  would  2  of  L.  In  a module w i t h  ••/  Primal analogous  (RCC),  cr P.  Conversely,i f r e s i d u a l s o f L, t h e n  residual  •  a p r i m a l submodule L h a s a u n i q u e  r e s i d u a l J? w h i c h  i s prime.  We c a l l  submodules allow; d e c o m p o s i t i o n  maximum  L P-primal.  and u n i q u e n e s s  theorems  t o those f o r t e r t i a r y , s u b m o d u l e s .  Definition  36:  intersection  I f a s u b m o d u l e L c a n be e x p r e s s e d as a f i n i t e o f s u b m o d u l e s , L = L^  the i n t e r s e c t i o n  Lemma 3 7 :  2  r e s i d u a l o f L^ w h i c h  then t h e i n t e r s e c t i o n next  L  n • • • n 1^, s u c h  that  i s i r r e d u n d a n t a n d no submodule L^ c a n be  r e p l a c e d by a r i g h t  The  0  be c o n t a i n e d ' i n a m a x i m a l p r o p e r l e f t  J |  proper l e f t  + A  Hence,  (L+r-Pg) = L - i - ^ + P g ) - L  2  or  and ^  r e s i d u a l P.  i s called  strictly  c o n t a i n s L^  reduced.  lemma i s t h e one r e f e r r e d t o f o l l o w i n g Theorem 2 8 .  I n a submodule w i t h  (RCC), e v e r y n o n - p r i m a l  L h a s a r e p r e s e n t a t i o n as a r e d u c e d i n t e r s e c t i o n r e s i d u a l s o f L one o f w h i c h  i s primal.  submodule  o f two r i g h t  30  Proof: L*-K  I f L i s not p r i m a l there  D L, L+-A  1  gives  2  z> L and L = (L-s-A.^  the ascending  chain  duals  let  o f L,  n (L-s-Ag).  c o n d i t i o n on l e f t  have t h e d e s c e n d i n g c h a i n show t h i s  e x i s t s A j , A^ s u c h  t h e (RCC)  r e s i d u a l s o f L we  c o n d i t i o n on r i g h t  a  Since  that  r e s i d u a l s o f L.  be a d e s c e n d i n g c h a i n o f r i g h t  = L-s—B^.  Consider  the ascending  To  resi-  chain of l e f t  residuals  L—»L, cr L — s L _ cr • • • • F o r some n , L—»L = L—s-L , ± — <. — n n • J. . = ••• . W e h a v e B , L , c L so B , c L — s L . = L — s L , b u t —n+1 n + 1 — —n + 1 — n+1 n L 3 L so B L cr L i s a c o n t r a d i c t i o n . T h u s , t h e r e e x i s t s n n +1 —n + 1 n — ' A  X  a right  A  residual  x  of L which i s minimal i n the set of r i g h t  r e s i d u a l s which p r o p e r l y c o n t a i n of r i g h t  L and L cr  h a s a m a x i m a l e l e m e n t L^.  (L "S—J )  w i t h L^-s—_B^ r> L  (^-s-jp  fl N  1  2  L =  The s e t  1  = H  1  set of r i g h t  2  and ( L - * - B ) n ^ 1  2  then  = 1^ so t h a t 1^ cr L  and  1  Thus L^ must be p r i m a l .  The  r e s i d u a l s Y o f L s a t i s f y i n g Y rs H^ and Y n L^ = L  a maximal element L  lemma.  I f L^ =» (L^-t—J3^) 0  and L ^ - s - J Z)  1  which i s a c o n t r a d i c t i o n .  This  c L-t-Ag.  r e s i d u a l s o f L o f t h e f o r m X when X p L-s—A^ a n d  L = X  has  A  so t h a t L = L^ n L  2  2  s a t i s f y i n g the  B  lemma may be u s e d t o p r o v e t h e f o l l o w i n g e x i s t e n c e  which i s g i v e n here without  theorem  proof.  o  Theorem 3 8 :  I f M h a s (RCC) t h e n e v e r y  reduced i n t e r s e c t i o n of a f i n i t e L = L  1  for  n ••• D L #  a l l i ^ j .  k  and i f L  i  submodule L f M. h a s a  number o f p r i m a l  i sj^-primal, ?  ±  submodules,  £ I! j # -Ej  £  31  Uniqueness  follows  Theorem 39: tions  I f M has  (RCC)  and L f M has  and t h e s e t s  uniquely defined  proper l e f t  D.  readily. two  reduced r e p r e s e n t a  as i n T h e o r e m J>&, t h e n t h e number o f c o m p o n e n t s i s t h e  same f o r b o t h The  quite  The  of primes  set of primes  residuals  of  coincide  i n some o r d e r .  i s the set of maximal  L.  r e l a t i o n s h i p between p r i m a r y , t e r t i a r y  and p r i m a l  sub-  modules. By Theorems 16, 25 primary, t e r t i a r y If  M has  (RCC)  (1) set  So  and p r i m a l  then  any  p r o p e r prime  of i d e a l s  and 35  containing  s u b m o d u l e L has left  residuals  (3)  maximal proper l e f t  L primary  residuals  <£r^.  L has  ideals  L  primal  which  <^z>  many  are minimal  i n the  illustrated  below:  and residuals.  we h a v e t h e r e l a t i o n s h i p s exactly  residual  L tertiary  finitely  residuals.  L—tM,  essential  (RCC)  can r e l a t e t h e d e f i n i t i o n s o f  submodules t o p r o p e r l e f t  (2)  i f M has  we  one p r o p e r p r i m e  left  and i t i s minimum i n t h e s e t o f  containing  L—tM.  L has  exactly  one  essential  L has  exactly  one  maximal p r o p e r  residual.  residual. left  32 PART I I I G e n e r a l i z a t i o n s  A»  o f t h e Second Uniqueness  Theorem  I s o l a t e d components.  1:  Definition  The r i g h t ideals prime  L e t A be an i d e a l upper  M component  B c o n t a i n i n g A such t o B.  By d e f i n i t i o n ,  Proof:  that  The n o t a t i o n  B  a  n  X i s such  Lemma 3 :  are sets  and  M c M 1  Proof: m e  and 2  then  we d e f i n e for  the right  as A i t s e l f .  In what Lemma 5 -  an i d e a l  such  every m e  that  B f o r which  0 ^  every  3  f o r A.  i s r p t o X,  1  H  0 A • t  2  that  l o w e r M component  J|  t(h,  If H  M  o f A as (x e R: mfix £ A  I f M » 0 we d e f i n e the r i g h t  lower  M component  M),  t h e c a s e where M = 0 i s always  F o r any i d e a l  every  and M an m-system.  We use t h e n o t a t i o n  follows  u ( A , M) c u ( B , M ) .  c o n t a i n i n g A such  L e t A be an i d e a l  some m e M).  i s defined  2  I f X i s an i d e a l  L:  of M i s right  u C A ^ ) C u(A, M ) .  i s r p t o X then  Definition  of a l l  containing £•  containing  m £ M i s r p t o X, t h e n  o f K.  i s u ( A , M).  M n B » 0 , then  d  I f X i s an i d e a l  If  e v e r y element  s e t , t h e component  u ( A , M) i s an i d e a l  If A c  Lemma 2;  of A i s the i n t e r s e c t i o n  I f M i s the n u l l  t o be A i t s e l f .  o f a ring; R and M a s u b s e t  A and m-system M, i(A,  obvious.  M) i s an i d e a l *  33  Proof: m^Rx cr A, 2 m  m zm R(x 1  2  Lemma 6:  I f x, y £ R  y  — —•  - y ) cr A.  F  o  r  M) t h e n f o r some s  o  m  e  m  £ M,  0  z £ R, m^zn^ £ M and  The r e m a i n d e r  i s obvious.  J|  F o r a n y i d e a l A and ra-system M, A c t ( A , M) cr u ( A , M ) .  Proof:  A cr -t(A, M) i s o b v i o u s .  A cr X and e v e r y m £ M i s rp t o X.  L e t X be an i d e a l  such  that  T h e n , mRx cr X. i m p l i e s x £ X.  Let  y £ .{.(A, M) so t h a t mRy c A f o r some m £ M, mRy cr A o X so  y  X.  E  ]  Theorem 7:  F o r any i d e a l A and m-system M,  (a)  u [ u ( A , M), Mj = u ( A , M)  (b)  u[f,(A, M ) , M] = u ( A , M)  (c)  *,[u(A, M ) , M] = u ( A , M).  Proof:  ( a ) u (A, M) ro A so Lemma 2 i m p l i e s  u [ u ( A , M), M] ro u ( A , M).  L e t X be s u c h t h a t A c X and m £ M  i m p l i e s m i s r p t o X, t h e n X ro u ( A , M) by d e f i n i t i o n is  s u c h an i d e a l .  that  Notice  the operation  closure  operation.  which  that  and u ( A , M)  Lemma 2, Lemma 6 a n d ( a ) i m p l y  c o r r e s p o n d s t o A t h e i d e a l u ( A , M) i s a  Since this  i s s o , ( b ) f o l l o w s f r o m Lemma 6.  We h a v e ,t[u(A, M), M] rp. u ( A , M) b u t by Lemma 6, t [ u ( A , M ) , M] c u [ u ( A , M), MJ = u ( A , M ) . Theorem 8:  I f R has the ascending chain  Jj condition  for ideals  t h e n f o r any i d e a l A and m-s y s t e m M, u ( A , M) = * ( A , M ) . This  i s a result  o f B a r n e s * t o be f o u n d i n [ 1 2 ] (Theorem 1 2 ) .  34  The  remainder  generalization ponents  of t h i s  section  of the second  will  be t o g i v e t h e a c t u a l  uniqueness  t h e o r e m u s i n g t h e com-  d e f i n e d above.  If Q i s right  Lemma 9:  complement  of r ( ^ ) , then  If A =  10:  Q = u ( ^ , M).  f)  0  0 -2  i  x t P^ f o r i = 1, condition  r.  s  a  irredundant primary  n  k  o f A w i t h P^ = r ( ( ^ ) ,  representation  chain  M an m-system c o n t a i n e d i n t h e  Q cr .£ and e v e r y m e M i s r p t o Q.  Proof:  Theorem  primary,  then  x i s rp to A i f  I f the r i n g has t h e ascending  f o r ideals  the converse  also holds  ([ll]#  Theorem 1 2 ) . Theorem  11:  (The S e c o n d U n i q u e n e s s  fl^g A  A = tion  of A with  contains  U  (A,  a  i  P,.  n  irredundant primary representaI f P i s a p r o p e r prime  ideal  P but does n o t c o n t a i n P ..., — r r+l  which  P,, t h e n —K  I f P D P, t h e n u ( A , _P) cr u ( A , ^ . ) by Lemma 3.  i  2 A and e v e r y  element  of  Hence, u ( A , ^ ) cr £  ^P.) cr ^P) = A .  k, t h e n  follows  i s rp to j ^ . ±  Thus,  0 ••• .fl 5 -  I f r = k,  r  I f r < k and P does n o t c o n t a i n P j ,  j = r+1, This  e  r ( ^ ) = I\.  —1  Proof:  u(A,  b  Let  ^P) = £x n • • • n Q r -  U(A,  But, ^  A -^k  Theorem)  since  P does not c o n t a i n Q^  t  P. i s t h e u n i q u e  j = r + 1,  k.  c o n t a i n i n g Q..  minimum p r i m e  J  Thus, t h e r e m. t P. l — x  1 #  are elements  J  m^, m^, • • « , ™ _ k  r  £ Q r+i*  with  The e l e m e n t s ra, a r e i n t h e m-system ^JP so t h e r e i ~"  «.., x  k  r  _  1  £ R such  that  m = n^x^n^x.^  x k  _  r  _ i  m k  _  exist i  r  s  a  n  35  e l e m e n t o f ,^P. q £ ^  B u t , m £ £ + i 0 ••• D .Q » T  fl ••• fl jJ #  m R <  J £ A.  follows.  Thus  i f  I f X i s an i d e a l , A r X, and n t P fl ••• (1 J2 ,  i m p l i e s n i s r p t o X, t h e n q £ i m p l i e s q £ X.  So,  k  mRq c A c X and t h i s  R  p ••• fl ^  f  c u ( A , ^ P ) and t h e t h e o r e m  J|  This i s not t h e form o f t h e second uniqueness theorem Part  Zl>  I but i f { P  •••» 2  h  1 #  • • P , } i s an i s o l a t e d s e t o f p r i m e s l e t  be t h e m a x i m a l  primes i n the s e t .  u ( A , ^ £ ) fl ••• nu(A, ^ £ ) = .P^ D ••• fl Q. 1  h  Corollary  12:  and i f A = ^ if  P M  Then  i s uniquely  I f R has the ascending chain c o n d i t i o n fl ••• D .Q  k  13:  jQ^,  (RCC).  L e t R be a r i n g and M a l e f t  R-module w i t h  T h e n , f o r any submodule L and any p r i m e i d e a l the  f o r ideals  i s a m i n i m a l p r i m e c o n t a i n i n g A, t h e n u ( A , ^ P ) i s  A g e n e r a l component f o r m o d u l e s w i t h  Definition  determined.  i s a s h o r t p r i m a r y r e p r e s e n t a t i o n and  r i g h t p r i m a r y and i s e q u a l t o one o f t h e i d e a l s B.  given i n  P H  we  (RCC). define  i s o l a t e d .P component o f L, L p , as t h e u n i o n o f a l l s u b -  modules o f t h e form The  s e t o f submodules  the  (RCC).  L-t—B  when _B gz P.  D e f i n e L^ = L.  L-s—B where _B c£ P h a s a m a x i m a l  I f L-s—JB^ and L-s—B^  a r e two m a x i m a l  e l e m e n t by  elements  (L-s—JB^) + ( L - s — B ^ ) CZ L - s — j ^ j ^ w h i c h i s a member o f t h e s e t s i n c e P i s prime. The  Thus Lp i s w e l l (RCC) c o n d i t i o n  defined.  i s assumed t h r o u g h o u t t h e whole  and i s f r e q u e n t l y m e n t i o n e d  by i n s e r t i n g t h e s y m b o l  s t a r t of t h e statement of a theorem.  section  (RCC) a t t h e  36  Lemma 14:  I f L« = L t - B , B £ P, t h e n BL» c L cr N so  Proof:  3  L» Q N t - B c: N . p  Lemma 15 : L,  I f L ez N a n d P i s p r i m e t h e n Lp c Np»  (RCC).  I f P^ e~ Pg a r e p r i m e s , t h e n f o r any  (RCC).  Lp 2 Lp • - 1 - 2 Proof:  Lemma 1 6 :  I f L» = L t - B , B £ ? (RCC).  t  h  e  [Lp] 3 L p  I f N = Lp-t-C, C £ P we w i s h t o show t h a t C' £ P.  Now, CN cr L  N cr L-t—C*C ez L  p  n  d  L  * S p • B L  p  by Lemma 1 4 .  N cr L-t-C* f o r some  so f o r some C* £ P, CN ez L t - C * a n d  where C*C £ P s i n c e  p  a  F o r any L a n d p r i m e P, [ L p ] p = L p .  C l e a r l y Lp 3 L so t h a t  Proof:  J £ -F-i  n  2  3  P i s prime.  We may combine t h e above lemmas t o s t a t e T h e o r e m 17:  (RCC).  The f u n c t i o n  f i x e d prime P i s a c l o s u r e  w h i c h maps L o n t o Lp f o r any  operation  on t h e l a t t i c e  of sub-  m o d u l e s o f M. A further property Theorem 1 8 :  of t h i s  (RCC).  function i s  The f u n c t i o n  w h i c h maps L o n t o L  f i x e d p r i m e P i s an f l - e n d o m o r p h i s m o f t h e l a t t i c e  f o r any  p  of sub-  m o d u l e s o f M. Proof: LflNcrL, L t  We must show t h a t  p  f) N  L n N g N so by Lemma 14,  £ L , N« ez N p  L  p  p  - ( L fl N ) . We h a v e p  ( L n N ) ez L p  p  n N . p  t h e n f o r some B £ P, C £ P, BL* ez L and  If  37  Thus L» n H » cr ( L n M ) . J |  Then BC cr (L* n N * ) Q L fl N.  CN» cr N.  P  We a r e now a b l e t o c o n s i d e r t h e s e c o n d u n i q u e n e s s t h e o r e m i n the  more g e n e r a l s e t t i n g  Lemma 19:  o f modules.  I f M h a s (RCC) and Q i s p r i m a r y and i f P i s a p r i m e  such t h a t  r ( Q ) cr P t h e n Q  Proof:  = Q.  p  I f S £ P t h e n S £ r ( Q ) a n d i f L = Q-j-S t h e n  S L cr Q, S £ r ( Q ) s o L c Q. Theorem 2 0 :  So Q  c Q.  p  I f M h a s (RCC) and L = Q  1  ]  . 0 Qj, i s an i r r e d u n -  n •«•  d a n t p r i m a r y r e p r e s e n t a t i o n o f L w i t h r ( Q ^ ) = P^ and i f ? i s a proper prime  i d e a l w h i c h c o n t a i n s P^,  contain P —r+1*Proof: L  cz Q  p  i p  n  0 Q  Q. n • " • fl Q • —i n P -> — r +1 r  Q  1  r  ^i^» ^ - n _ P— r in +2  n •••  P / M C  Q..  P ^  1  If  [P^,  r  P  1  of  p  o  m  e  n  i»  n. £ Q^*  i p  0 ••• 0 Q  ("I ••• P Q  P  k k  £ L . p  (Q n i  This i s containedi n  •••  k  since  n Q ) £ , L and r  k  £ P since P i s prime.  fe  P j } i s an i s o l a t e d  Hence Q  1  fl ••• D Q  r  £ L .  s e t o f p r i m e s , l e t P^,  p  P^  p r i m e s o f t h i s s e t . Thus L fl ••• 0 L • -1 -h p  fl ••• fl Qj w h i c h i s t h e s e c o n d u n i q u e n e s s t h e o r e m Part I .  Since  .— - --  Consider  r+  1  r  =  f P  "  a n d i s c o n t a i n e d i n Q ^ fi ••• 0. Q  p  be t h e m a x i m a l Q  s  Thus P ^ * n  0 • • • 0 Qj, so t h a t  p  We n e e d t o show t h a t  n, V—k( « i 1 0 ••• 0 .Q^ r) •  0 Q  +1  N  o r  = Q^  p  and by Lemma 1 9 , Q  r P  r  =  b u t does n o t  f  P, , t h e n L = Q, p • • • f> Q • —k P 1 r  By Theorem 18, L  n  P  p  i n t h e form  J  38  Theorem 2 1 : primary  I f M h a s (RCC) and L •  representation  (1 ••• (1 Q  i s a short  FC  o f L, t h e n a p r i m e P c o n t a i n i n g  L—tM i s one o f t h e p r i m e s P^ = r ( Q ^ ) , i = 1, • only  i f , Lp*—P 3 L . p  Proof: it  r ( L ) = r ( Q ) n • • • Tl r ( Q ) = ?  Since  i s clear  that  1  p r + 1  *  Then L  fc  = 0^ n . . .- n Q  and t h i s  ; P  r  isa  Lp*—A 3 Lp i f , and o n l y i f ,  of Lp,  I I ) and, t h e r e -  Lp+-P 3 L p . (ii)  Assume P 3 L—sM i s a p r i m e and P i s s u c h  Lp-s—JP 3 L p .  Since  that  t h e m i n i m a l p r i m e s c o n t a i n i n g L — t M a r e among  P^, ••», P^, P must c o n t a i n one o f them so assume  1 2. P f "••# l  a  z  X  is  p  P j ez P f o r some j , 1 < j < r (Lemma 18, P a r t  fore,  the  fl ••• f! P  1  Then P 3 P^, •  ( i ) Let P = P .  short primary representation A c  fc  t h e p r i m e s P^, •'"*/ Pj. i n c l u d e t h e m i n i m a l  primes c o n t a i n i n g L—tM. with P ±  k, i f , a n d  n  d  P ± lT+1>  •"•#  a short primary representation  Lemma 18 o f P a r t  I I gives  As an i m m e d i a t e c o r o l l a r y first  that  P -  T  k  h  e  n  L  p  = Q  of Lp. S i n c e  x  n ••• 0 Q  r  Lp-t—P 3 L p ,  f o r some i , 1 < i < r , P ez P^.  we h a v e Theorem 20 o f P a r t  I I orthe  uniqueness theorem f o r primary r e p r e s e n t a t i o n s  since the  prime r a d i c a l s particular  i n a short  representation  a r e n o t d e p e n d e n t on t h e  p r i m a r y modules o f t h e r e p r e s e n t a t i o n .  The p r o o f s multiplicative  above can be g e n e r a l i z e d lattices  immediately t othe  o f [ 9 ] and i n p a r t i c u l a r  t o t h e case  where L, N, ••• and A, JB, ••• a r e a l l t w o - s i d e d i d e a l s same r i n g .  With the obvious n o t a t i o n  ]  we h a v e  ofthe  39  Theorem 2 2 : prime  F o r any r i n g R w i t h  i d e a l P, A  Proof:  p  ( R C C ) , and any i d e a l A and  = ^ ( A , ,^P). t h e n f o r some m t P, a R x c A.  I f x e -l(A, ^ )  raR(x) gz A and (m) ( x ) cr A, (m) £ P so ( x ) cr A x e A [x:  p  t h e n f o r some C £ P, x e A t — C .  But,  and x e A .  p  2  The n e x t s t e p w o u l d be t o f i n d r e p r e s e n t a t i o n s , however i n t h i s i s a uniqueness theorem  Lemma 2 3 : is  I n a module w i t h  a reduced primal  maximum p r o p e r l e f t primal  analogous r e s u l t s f o r t e r t i a r y  c a s e much l e s s may be s t a t e d .  f o rprimal  analogous t o t h e second uniqueness  representations  (RCC), i f L = ^  representation  fl ••• p. L^ (k > 2 )  o f L and i f t h e u n i q u e  r e s i d u a l o f L, i s P. t h e n L_ i s P.— l — i P^ — l  p  Proof:  residual L = L Lp L  We w i l l p r o v e t h e r e s u l t f o r i = 1.  2  fl ••• fl L^.  i s contained  p +-A rr> L - 1 - 1  p  .  L  i n P^.  i+1  By Theorem 18,  P^ i s t h e maximum p r o p e r  = L^.  Thus L cr L  p  Next, every proper l e f t  L^  c  t  left  and  r e s i d u a l of  I f n o t , we h a v e A £ P^ and  Then f o r some B £ P, , B A ( L +-A) cr L, BA £ P -1  s i n c e P, i s p r i m e 1  Since  o f L^, we h a v e L^p fl L  p  n  p  n ••• fl L^p .  = L^p  which i s  theorem.  and L c L,, L = ^ fl ••• fl fl L —i — i fl ••• fl L^ i s a r e d u c e d p r i m a l r e p r e s e n t a t i o n .  p  niRx cr A and  t  x e -t(A, ^ P ) .  L  =  Cx <- A J cr ( x : .CRx cr A} SO f o r some m e C fl ~£_  There  If  p  A-J—C  Hence,  p  and so L  p  -s—A cr L which - 1 - 1 p  1  i s a contradiction.  40  Thus A £  V  i m p l i e s L -s-A = L . -1 -1 p  maximal p r o p e r l e f t  L  p L  p  L  p  1  .  Hence, L  We must now  p  1  show t h a t  L = Lp  n  L^.  D *•• H ^  by a r i g h t  residual  L-s—P  s  2  0 (Lg-s—A) f| ••• fl L^  which  r e s i d u a l o f L.  i s reduced.  r e s i d u a l which i s  i s P g - p r i r a a l , A cr .£  Since  2  2  a  n  d  i s a contradiction  Similarly  f o r any  J|  C o r o l l a r y 24:  I n a module w i t h  a reduced p r i m a l  in  (1 ^  So assume L = Lp  s i n c e Pg i s a p r o p e r l e f t  Lp  and P^ i s a p r o p e r l e f t  L = Lp  (Lg-s—P ) fi ••• fl Lj,  component.  0  i-R^  p  this,  i s P,-primal.  l a r g e r t h a n L^.  where L^*—A 3  To show  p  Lp  R e p l a c e one c o m p o n e n t , s a y L^, strictly  r e s i d u a l of L . -1  = L cr L+-P-J = ( L  fc  f| ••• fl Lj, so Lp * — P j of  By t h e a b o v e , i t i s enough .  p  n ••• 0 L  2  1  r e s i d u a l of L . -1  t o show t h a t P. i s a p r o p e r l e f t note t h a t  We n e x t show t h a t P, i s a  p  (RCC),  representation  i f L = L^ f)  fl L^ i s  o f L where L^ i s P ^ - p r i m a l ,  i s t h e s m a l l e s t j ^ . - p r i m a l s u b m o d u l e w h i c h may  replace  L.  the r e p r e s e n t a t i o n .  By a p p l y i n g t h e method o f Lemma 23 k t i m e s we h a v e Theorem 2 5 :  I n a module w i t h  maximal proper l e f t  (RCC) i f L i s a s u b m o d u l e w i t h  r e s i d u a l s P_^,  P^, k > 1, t h e n  L = L 0 L Q ••• n L i s a reduced p r i m a l -1 -2 -k p  p  p  representation  If  with  n  i s any r e d u c e d p r i m a l  t h e components l a b e l l e d so t h a t  - i This  n  L =  representation  1^ i s P ^ - p r i r a a l t h e n  1  i s a uniqueness theorem r e p l a c i n g the second uniqueness  theorem f o r p r i m a l  decomposition.  Lemma 23 may be weakened somewhat as f o l l o w s . Theorem 26; if  I f M h a s (RCC)  P i s an e s s e n t i a l r e s i d u a l o f L, Lp i s P - p r i m a l . Proof:  - P.  x  ~  - T  1  Now, L ^ = T p  n T  intersection  i p  ^  n  T  2 p  ^  n  and  k p  ~  1  ••• fl T  k p  ^.  x  Hence,  i s not redundant i n t h i s k  2 (~\  s i n c e i f i t were,  B u t , P_ i s t h e  «= T^.  k  1  tertiary  i s P ^ - t e r t i a r y and t h a t  r e s i d u a l o f T^ s o T^p  f) ••• fl T  2 p  ~  1  be a r e d u c e d  fc  o f L a n d assume t h a t  maximum p r o p e r l e f t Lp  fl • • • 0. T  Let L -  representation P  and L i s a submodule o f M t h e n  T.  i=2  p  2 f^] T^ i 2  —1  a  which c o n t r a d i c t s the irredundancy  of the representation  Hence, t h e r e  representation  i s a reduced t e r t i a r y  o f L.  of L with -1 a P ^ - t e r t i a r y component. T h a t i s , P^ i s an e s s e n t i a l r e s i d u a l o f L . B u t , A £ P., i m p l i e s L -j—A = L so P, i s t h e maximum -1 -1 -1 p  p  p  1  proper l e f t  p  1  r e s i d u a l of L . -1 p  Therefore,  L i s P.-primal. -1 p  1  ]|  U2  PART IV  Examples  Example  1;  The  and Remarks  (RCC) c o n d i t i o n  the e x i s t e n c e o f p r i m a r y  representations.  ( [ 2 ] ) i s a noncommutative Let e  l '  e  2'  K be  n  w  e^n  ne  (e ),  (  1  and  ^  e  e  i  e  =  e  ) ' ^ ^'  ^ i '  n  2  ( )»  e  e  The m i n i m a l  e  2  e;L  e  )  n  2  =  e  2*  n  ^  2^  a  n  T  primes  h  e  e  2  is (e )  P  e  r  i  =  e  m  e  condition. by  ^'  =  o f $ are c l e a r l y ( 0 ) ,  e  s  a  r  ( i»  e  e  e 2  )» ^  i)  e  ( e ) so t h a t 2  (n) so (n) i s n o t  (0) i s not p r i m a r y .  fl-irreducible'and  and  But,  so does n o t have  representation.  tertiary  example  K generated  2 l  c  Hence  a primary  }  e  But, (e^ie^  (e ) = (n).  (0) i s , l' 2  l 2  a r e ( e ^ ) and  prime.  (  e  The i d e a l s  d  guarantee  the ascending chain  ®'  =  to  The f o l l o w i n g  and § t h e a l g e b r a o v e r  i *  e  r i n g with  ,n = ne, = 0.  = n,  n  r(0) = (  r  a field  i s not s u f f i c i e n t  (e ) *  The  ideal  essential  s  2  (e )  x  ?  residual  of  (O)-*^)  (n)  (0) s i n c e  = {x: x (  (O)-e-(n) =  e i  ) = 0}  [ x : x ( n ) = 0}  = ( e ) and 2  = (e ). 2  (0)  To e m p h a s i z e t h e d i f f e r e n c e ideals which  the f o l l o w i n g i s not  Example  2;  1 a b c d e f  is a finite  between p r i m a r y example w i t h  and  tertiary  a tertiary  ideal  primary. Consider the r i n g  0 f d d b b a  f 0 d e b c 1  d d 0 f a 1 c  d e f 0 1 a b  b b a 1 0 f e  b c 1 a f 0 d  g i v e n by t h e f o l l o w i n g  a 1 c b e d 0  a b c d e f  0 a 0 a 0 a  0 b 0 b 0 b  a a c 0 e e  0 d 0 d 0 d  a 0 c a e c  tables:  a d c b e 1  (0)  43  T h a t t h i s i s a r i n g i s shown by t h e f a c t  that  satisfied  a field  two.  a -  d =  The  by t h e f o l l o w i n g  The z e r o  and 1 e l e m e n t s a r e t h e z e r o  /0 I 0 \0  0 0 0  1\ 0 ) , 0/  / 1 0  0 0  1\ 0 ,  Vo  /1 b = I O \0  e  = 0  0 0 0  / 0  o o/  two-sided  m a t r i c e s over  0\ 0 J 0/  are:  0  of c h a r a c t e r i s t i c  and i d e n t i t y m a t r i c e s .  c =  0 0 \ 1 0 ) ,  \0  ideals  ,  the tables are  / 1 0 f = ( 0 1  1 /  \0  1 \ 0 ) . 0  1./  ( 0 ) , ( a ) = [0, a } , ( b ) = ( d ) - ( a , b) =  {0, a, b, d},' ( c ) = ( e ) - ( a , c ) = fO, a, c , e }  f  R.  2  Since  R has a u n i t , R  • R, R ( a , b) =  ( a , b)R = ( a , b) so ( a , c ) i s p r i m e . (a,b)^  ^)(a,c)  Similarly, \(0)  ( a , b) i s p r i m e .  To compute  we need t h e maximum i d e a l A  t h a t ((O)-t-A) n X = (0) i m p l i e s But,  Since  and (0)-s—(a, c ) = ( x : ax = c x = e x = o } = ( a , b ) .  (O)-j-R = (0) and ( O ) - j - ( a , = ( a , c ) . To f i n d \ ( ( a ) )  b) = { x : ax = bx = dx = 0} » (0), we n o t e  ( a , c ) n ( a , b) = ( a ) and ( a , b) £ ( a ) . 1  and  X = (0).  (O)-t-(a) = { x : ( a ) x = 0} = ( a , b) w h i c h s a t i s f i e s t h e  requirement  X(0)  such  ( a , b) fl  ( a , c ) = ( a ) as b e f o r e  ( a ) - * - ( a , b) = ( a , c ) b u t Also,  ( a ) t - ( a , c) = (a,b)  and ( a , c ) £ ( a ) .  Finally,  then A.((a)) = ( a ) . The r((a))  primary  = (a).  radicals  Since  (a) i s not prime,  T h i s e x a m p l e may a l s o We h a v e (cj  can be s e e n t o be r ( 0 ) = ( a ) and (0) i s n o t p r i m a r y .  be c o n s i d e r e d as a l e f t  R-module.  ( a | = [0, a } , ( b | = ( a , b| - ( d | = {0,a,b,d},  = [0, c } , ( a , c| = {0, a, e, e } , ( e j =  [Q,  e } , R.  I t can  44  be  = ( a , c ) , ?vj((a|) • ( a ) , r ( 0 ) = ( a ) and  shown t h a t \(0)  r((a|) - (a). Example 3: ascending  The f o l l o w i n g e x a m p l e i s a c o m m u t a t i v e r i n g chain  c o n d i t i o n i n which there  are n o t p r i m a r y . three  2  K.  which  ring in  Consider  the i d e a l  2  I = ( x y, x y ) .  This  2  (x, y ) ( x y ) c contained  ideal  i s not primary  since  2  ( x y, xy ) and ( x y )  I and no power o f ( x , y ) i s  i n I . .However, i t i s e a s y t o s e e t h a t  maximum p r o p e r l e f t Remark 4:  residual  The t e r t i a r y r a d i c a l may be d e f i n e d  s e e n t o be e q u i v a l e n t  obviously  The d e f i n i t i o n s t o t h e one g i v e n  a r e w e l l d e f i n e d even i f t h e r e  Definition i  In a r i n g  ( x , y) i s t h e  of I .  elements o f a r i n g o r module. easily  are p r i m a l i d e a l s  L e t K [ x , y , z ] be t h e p o l y n o m i a l  commutative v a r i a b l e s over a f i e l d  with  i n terras o f given i n Part  below are I I but  i s no (RCC) c o n d i t i o n .  R, t h e t e r t i a r y r a d i c a l  of a l e f t  ideal I  i s t h e s e t o f elements a £ R such t h a t b % I i m p l i e s t h e r e x £ (b| such t h a t x t  I and aR*x cz I ( a R * x = aRx (j [ a x } ) .  I n a r i n g R, t h e t e r t i a r y r a d i c a l is  of a (two-sided)  the s e t of elements a £ R such t h a t h t  exists  x £ (b) such t h a t  Any  two-sided  considered Since  x $ I  ideal  ( b | cz ( b ) t h e r a d i c a l  the  radical  the  two r a d i c a l s  i d e a l _I  there  and aR*x cz I»  and when c o n s i d e r e d of the l e f t  ideal  —  one when  as an i d e a l . i s contained i n p . 469)  that  c o i n c i d e i f t h e r i n g has the d e s c e n d i n g  chain  of the i d e a l .  c o n d i t i o n on l e f t  I implies  i d e a l h a s two t e r t i a r y r a d i c a l s  as a l e f t  exists  ideals.  I t can be p r o v e d  ([6],  45  Remark 5 i  Other c o n n e c t i o n s  between  s t r u c t u r e s have n o t been f u l l y M is a left ideals sided  R-module w i t h  of R a l s o ideal  simple  L—tM have  two-sided  the i d e a l  the  residual  representations.  R-module w i t h  o f R has  L.  i s a proper  cz L.  left  o f L.  The  following  left  left  of  residual  r e s i d u a l of  ez A.(L).  L, a p r o p e r  of a l l i d e a l s  left  C such  L = L—tM, AM £  that  L and  Hence, _B = _L—tA c  L—tAM.  Under t h e a s s u m p t i o n s o f t h e theorem, i f  L = T^ 0 ••• fl Ij, i s a t e r t i a r y i f L—tM = T^ p  and  X(L—*M)  But, s i n c e A £  residual  L and t h e two-  any p r o p e r  i n a proper  i d e a l _B i s t h e sum  of two-sided  (RCC) and t h e l a t t i c e  (RCC), t h e n  In p a r t i c u l a r ,  The  an example, i f  between t h e two.  L e t L—tM = L and B = L — t A , A £  of _L.  Corollary:  a submodule  some r e l a t i o n s h i p s  L = L—tM o r CAM  L—tAM  both  L—*M i s c o n t a i n e d  submodule  Proof:  CAc  ideals  As  and module  (RCC) and i f t h e l a t t i c e  tertiary  If M i s a l e f t  ideal  investigated.  (RCC) then  theorem g i v e s  Theorem:  of  has  related  •••  representation  0 T^ i s a t e r t i a r y  w i t h A-(T^) = P^  representation  with  A.(T.) = P! t h e n each P! cz P. f o r some j . - i - i - i j Remark 6 : are  In t h i s  remark some w e l l - k n o w n p r o p e r t i e s  reworded to c o r r e s p o n d to the language of t h i s E v e r y maximal  ideals  are c l e a r l y  left  ideal  is tertiary  P-irreducible.  c o n s i d e r X,(T) = {a: b & T i m p l i e s that  x t  clearly  T and aR*x c TJ and T-*R a £ A.(T).  s i n c e maximal  exists  b I  left  left  ideal,  x £ ( b | such  = f a : aR ez T J .  I f a £ A.(T) f o r e v e r y  rings  thesis.  I f T i s a maximal there  of  I f a e T—tR,  T there  i s an  3  46  x t T and  x e (b| such t h a t a(T + ( x | ) = aR that  T—s-R i s t r i v i a l l y  no i n t e r m e d i a t e I,  left  every e s s e n t i a l  r e q u i r e d ) and It R/P  i s well  ideal.  a e T—sR.  residual  known t h a t ring)  maximal l e f t  I—s-R = 0.  tertiary  i s prime  Part  i f i t has  a maximal l e f t  Thus a r i n g i s p r i m i t i v e  r a d i c a l o f some m a x i m a l l e f t  chain condition those p o s s i b l e  on  w i t h the  not yet a v a i l a b l e ideals The  left  in this  residual  o f [9]»  ideals. (RCC)  ideal  I such  i f the zero i d e a l i s the  Lesieur  and C r o i s o t w i t h the  These r e s u l t s  condition.  g e n e r a l theorem  have  Similar  results  w i t h maximum  given  descending  are s t r o n g e r  o r i n t h e more g e n e r a l c a s e o f f i n i t e  than  are  condition dimensional  gives r i s e to a simple  direction.  Let R have  P-tertiary  maximal  ideal.  f o r the case of r i n g s  following  (i.e.,  i d e a l and so i s p r i m e .  t e r t i a r y decompositions in rings  Theorem:  are  is  in R  i f i t i s the e s s e n t i a l  r e s u l t s on  result  there  By Theorem 14,  an i d e a l P i s p r i m i t i v e  In C h a p t e r V I I I  modules.  of T since  (no c h a i n c o n d i t i o n  Remark 7:  on l e f t  Note  i f P = L—*R where L i s a m o d u l a r  A ring R i s primitive that  residual  between T and R.  Thus P i s p r i m i t i v e  a modular  Hence X,(T) = T—*R.  an e s s e n t i a l  ideals  hence  so T—s-R i s p r i m e .  i s a primitive  left of  T and  c  aR*x c T so a R * ( x | cz T,  (RCC)  as a l e f t  t h e n a £ R such t h a t  R-module and  i f T is  Ra c£ T i m p l i e s  T—sa i s  P - t e r t i ary. Proof:  The  residual  T—ta i s c l e a r l y a l e f t  ideal.  We  show  47  first  t h a t A.(T—ta) = P.  There e x i s t s x £ (ba|  f  x £ ( b a | such t h a t  x t T and r ( x ) ez T.  x = c a f o r some c £ ( b | .  ez T—»a.  r(c|  L e t r £ P and b t T—*a, t h e n ba t T.  Hence r £ A . ( T — t a ) .  Thus c t T — t a and  Conversely, i f r £  t h e n s i n c e Ra £ T, t h e r e i s some x t T — t a . some x* £ ( x | s u c h t h a t  r ( x | cz T — t a .  i s t h e maximum p r o p e r  b £ P = 7v.(T-ta).  if and  f  corollary  b £ P.  ( i t i s true  as a l e f t Proof: n  also i n the  on l e f t  ideals  R-module  and R h a s t h e  then i f T i s P - t e r t i a r y  and  i d e a l s u c h t h a t RL £ T t h e n T — t L i s P - t e r t i a r y ideal.  L is finitely If  i=l the i n t e r s e c t i o n  g e n e r a t e d so L = ( a , , *'*,  J |  a | and  Ra, ez T t h e n T — t a , = R i s s u p e r f l u o u s i n  so we may  assume t h a t  t h e o r e m e a c h T—ta^^^ i s P - t e r t i a r y . the r e s u l t .  that  [9])«  I f R i s c o n s i d e r e d as a l e f t  left  It follows  J  i s immediate  condition  maximum c o n d i t i o n L i s a  implies  i d e a l s , B ± T — t a , AB ez T — t a t h e n A cz P  so T — t a i s P - t e r t i a r y .  Corollary:  P  I f b ( c | ez T — t a , c t T — t a t h e n c a i T and  A and B a r e l e f t  c a s e o f minimum  (since  o f T so r £ T — t ( x a | ez P.  residual  cz T, b u t s i n c e T i s P - t e r t i a r y ,  The f o l l o w i n g  Hence, t h e r e i s  But, since T i s P - t e r t i a r y ,  show t h a t b ( c | c T — t a , c t T — t a  We n e x t  b(ca|  left  \(T—ta)  Then r ( x * a | cz T  f  r ( x * a | Q r ( x * | s ) and x»a t T.  Since  Ra^ £ T and so by t h e  Thus Lemma 26 o f P a r t I I g i v e s  48  References 1.  2.  W. E. B a r n e s . P r i m a l i d e a l s and i s o l a t e d components i n nonc o m m u t a t i v e r i n g s . T r a n s . Amer. Math. S o c , 8J2 ( 1 9 5 6 ) , 1-16. C. W.  C u r t i s , .On a d d i t i v e  ideal  Amer. J o u r , o f Math., 1 4 ( 1 9 5 2 ) ,  theory  in general  687-700.  rings,  3.  L. F u c h s , .On p r i m a l  4.  A. W. 1961,  5«  L. L e s i e u r and R. C r o i s o t , T h e o r i e n o e t h e r i e n n e des anneaux, des d e m i - g r o u p e s e t des modules dans l e c a s non c o m m u t a t i f I, C o l l . d ' A l g . Sup., C.B.R.M. (1956"), 7 9 - 1 2 1 .  1-8.  ideals,  G o l d i e , R i n g s w i t h maximum c o n d i t i o n , mimeographed n o t e s ) .  6.  , Ibid.,  7.  I I , Math. A n n a l e n ,  , Ibid. . I l l ,  1 (1950),  P r o c . Amer. Math. S o c ,  Acad. Royale  134  (Yale  University,  (1958), 458-476.  de B e l g i q u e , .44  (1958),  75-938.  , La n o t i o n  de r e s i d u a l  essentiel,  Comptes  Rendus  A c a d . S c . , .246 (1958), 3 5 7 - 3 6 0 . 9. 10.  Algebre N, H,  McCoy, P r i m e i de a l s  Math., 21 11.  noetherienne  non commut at i ve  in general  rings,  (Paris,  Amer. J o u r , o f  (1949), 823-833-  D. C. Murdoch, Cont f i but i on s t o noncommut at i ve i d e a l  Can. J o u r , o f Math., .4 ( 1 9 5 2 ) , 12. • with  1963).  43-57.  theory,  , S u b r i n g s o f t h e maximal r i n g o f q u o t i e n t s a s s o c i a t e d c l o s u r e o p e r a t i o n s . Can. J o u r . o f Math., .JJ5 ( 1 9 6 3 ) ,  723-743.  

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