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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 of B r i t i s h Columbia, 1959 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF Master of A r t s i n the Department of Mathemat i cs We accept t h i s t h e s i s as c o n f o r m i n g t o the r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH A p r i l , 1964 COLUMBIA I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at 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 , I agree that 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 study* I f u r t h e r agree that 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 of t h i s t h e s i s for . s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . I t i s unders tood that c o p y i n g or p u b l i -c a t i o n of t h i s t h e s i s 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 without my w r i t t e n p e r m i s s i o n * Department of ^ V ^ f i u ^ v ^ V i - c-C 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 , Vancouver 8, Canada Date i J U w J z ^ /CH ' i i A bst r a c t Two methods of g e n e r a l i z i n g the 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 to modules over a r b i t r a r y r i n g s are d e s c r i b e d i n d e t a i l . The f i r s t i s by e x t e n d i n g the p r i m a r y i d e a l s and i s o l a t e d components of Murdoch t o modules. The second i s by u s i n g the t e r t i a r y sub-modules of L e s i e u r and C r o i s o t . The development i s s e l f - c o n t a i n e d except f o r ele m e n t a r y n o t i o n s of r i n g and module t h e o r y . The d e f i n i t i o n of p r i m a l subraodules w i t h some r e s u l t s i s i n c l u d e d f o r completeness. Some c o n c r e t e examples are g i v e n as i l l u s t r a t i o n s . Acknowledgement I would l i k e to thank Dr. D. C. Murdoch f o r sugges t i n g the t o p i c and f o r t a k i n g time d u r i n g an e s p e c i a l l y busy year to supe r v i s e my work. i i i T a b l e of Contents Page I n t r o d u c t i o n 1 N o t a t i o n 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 Theory and the Theory of R e s i d u a l s 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 P a r t I I . G e n e r a l i z a t i o n s of the Decomposition Theorem and the F i r s t Uniqueness Theorem 12 A. P r i m a r y I d e a l s and P r i m a r y Submodules 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 the Primary, T e r t i a r y and P r i m a l Submodules 31 P-art I I I . G e n e r a l i z a t i o n s of the Second Uniqueness Theorem 32 A. I s o l a t e d Components 32 B. A G e n e r a l Component f o r Modules w i t h (RCC) 35 P a r t IV. Examples and Remarks 42 R e f e r e n c e s 48 I n t r o d u c t i o n . Since 1950, many new r e s u l t s have been p u b l i s h e d g i v i n g g e n e r a l i z a t i o n s of the c l a s s i c a l Noetherian i d e a l theory to noncommutative r i n g s , modules over noncommutative r i n g s and m u l t i p l i c a t i v e l a t t i c e s . In t h i s t h e s i s I have presented the more important of these as a u n i f i e d t h e o r y . Noetherian theory may be expressed i n terms of the elements of the r i n g s and modules or i n terms of the a r i t h m e t i c of i d e a l s and submodules. The l a t t e r approach g e n e r a l i z e s to m u l t i p l i c a -t i v e l a t t i c e s . In most of t h i s t h e s i s I have r e s t r i c t e d a t t e n -t i o n to modules over a r b i t r a r y r i n g s and developed the theory u s i n g the a r i t h m e t i c of i d e a l s and submodules. Proofs which do not i n v o l v e the use of elements may be g e n e r a l i z e d to m u l t i p l i c a -t i v e l a t t i c e s merely by changing a few words. This t h e s i s i s d i v i d e d i n t o four p a r t s . Part I contains a b r i e f o u t l i n e of the c l a s s i c a l Noetherian theory. A l s o i n t h i s part i s an o u t l i n e of the theory of r e s i d -u a l s . R e s i d u a l s are g e n e r a l i z a t i o n s of the r e s i d u a l s of commuta-t i v e i d e a l theory ( i . e . , i f A and B are i d e a l s i n a commutative r i n g , the i d e a l A:B = [c:c_B c A} i s c a l l e d the r e s i d u a l of A by JB). It i s t h i s theory which allows us to avoid the use of elements i n l a t e r p a r t s of the t h e s i s . The two major g e n e r a l i z a t i o n s of primary i d e a l s — the primary submodules of Murdoch and the t e r t i a r y submodules of L e s i e u r and C r o i s o t — are d i s c u s s e d i n Part I I . The extensions of the '•existence of decompositions'* and the ^uniqueness of asso-c i a t e d primes 1* theorems of the c l a s s i c a l theory are given f o r both primary and t e r t i a r y submodules. The p a r t concludes with a b r i e f d e s c r i p t i o n of p r i m a l submodules. 2 The e x t e n s i o n of the "uniqueness of i s o l a t e d components'* theorem i s the t o p i c of P a r t I I I , The f i n a l p a r t c o n t a i n s some examples and remarks. The remarks i n d i c a t e some of the problems which need f u r t h e r i n v e s t i -g a t i o n . N o t a t i o n . Throughout t h i s t h e s i s the f o l l o w i n g n o t a t i o n w i l l be used. 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 r e p r e s e n t i d e a l s ('•two-sided' 1 u n d e r s t o o d ) . P l a i n c a p i t a l s L, M, ... w i l l r e p r e s e n t o n e - s i d e d i d e a l s or submodules. The symbols ( S ) , ( S|, |S) are the t w o - s i d e d , l e f t and r i g h t i d e a l s , r e s p e c t i v e l y , g e n e r a t e d by the set S. The symbols c and c are ,for s et i n c l u s i o n and p r o p e r s e t i n c l u s i o n r e s p e c t i v e l y . Set complements are i n d i c a t e d by ^. Sum, unions and i n t e r s e c t i o n s of submodules may be e x p r e s s e d by { ^ L s P } , [|jL:P], [nL:P| which may be read as "the sum of a l l sub-modules L h a v i n g p r o p e r t y P t t, e t c . 3 PART I C l a s s i c a l N o e t h e r i a n Theory and the Theory of R e s i d u a l s A. C l a s s i c a l N o e t h e r i a n Theory. N o e t h e r i a n i d e a l t h e o r y f o r commutative r i n g s 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 o u t -l i n e w i l l be g i v e n h e r e . An i d e a l P of a commutative r i n g R i s c a l l e d prime i f the pr o d u c t ab £ P and a t P i m p l i e s t h a t b £ P or i f AB rz P and A <£ P then B c P. An i d e a l Q i s c a l l e d p r i m a r y i f the product ab £ .Q and a t Q i m p l i e s t h a t b n £ Q f o r some p o s i t i v e i n t e g e r n. I f R has the a s c e n d i n g c h a i n c o n d i t i o n f o r i d e a l s then every i d e a l i s f i n i t e l y g e n e r a t e d so t h a t _Q i s p r i m a r y i f , and o n l y i f , JL§ £ .Q and A Q i m p l i e s Bn Q Q f o r some p o s i t i v e i n t e g e r n. The r a d i c a l r ( A ) of an i d e a l A i s d e f i n e d by r ( A ) = {r £ R: r n £ A f o r some p o s i t i v e i n t e g e r n}. The r a d i c a l of an i d e a l A i n a r i n g w i t h the 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 e x p r e s s e d as r ( A ) = {^B: B n c A f o r some p o s i t i v e i n t e g e r n}. The D e c o m p o s i t i o n Theorem; In a commutative r i n g w i t h the as c e n d i n g c h a i n c o n d i t i o n , e v e r y i d e a l can be e x p r e s s e d as the i n t e r s e c t i o n of f i n i t e l y many p r i m a r y i d e a l s . I f A = fl •"" 0 Qy, i s an i n t e r s e c t i o n as d e s c r i b e d i n the theorem, s u p e r f l u o u s terms may be d e l e t e d so t h a t the i n t e r -s e c t i o n i s i r r e d u n d a n t . I f two p r i m a r y i d e a l s have the same r a d i c a l P, t h e i r i n t e r s e c t i o n i s p r i m a r y w i t h r a d i c a l P. Hence, the i n t e r s e c t i o n d e s c r i b e d i n the theorem may be ar r a n g e d so t h a t the prime i d e a l s r(^j)» •••» r C 2 f c ) a r e d i s t i n c t ; i n t h i s case an i r r e d u n d a n t i n t e r s e c t i o n i s c a l l e d s h o r t . 4 The F i r s t Uniqueness Theorem; Let R be a commutative r i n g w i t h the a s c e n d i n g c h a i n c o n d i t i o n . Let A = f| ••" 0 .Qj, = .QJ 0 D .2£, be s h o r t r e p r e s e n t a t i o n s of an i d e a l A of R as i n t e r -s e c t i o n s of p r i m a r y i d e a l s . Then k = k f and the r a d i c a l s "'» r ( % ) a n d r ( 2 p f r ( g » t ) c o i n c i d e i n some o r d e r . I f A = ^  fl "•• fl 5 t i s a s h o r t r e p r e s e n t a t i o n of A as an i n t e r s e c t i o n of p r i m a r y i d e a l s and r C ^ ) = P 1 # r(-Q k) = P^; a subset of the primes P^, P.^  say, {P^ , P^ } i s c a l l e d \ i s o l a t e d i f P c P. i m p l i e s t h a t P i s one of the P. P. • The c o r r e s p o n d i n g i n t e r s e c t i o n Q, f) ••• Q i s c a l l e d an " l ' j i s o l a t e d component of A. The Second Uniqueness Theorem: Let R be a commutative r i n g w i t h the a s c e n d i n g c h a i n c o n d i t i o n . Let A <= • •• n Qy. be a s h o r t r e p r e s e n t a t i o n of an i d e a l A as an i n t e r s e c t i o n of p r i -mary i d e a l s . Then the i s o l a t e d components of A are u n i q u e l y d e t e r m i n e d ; t h a t i s , they are the same f o r a l l s h o r t r e p r e s e n t a -t i o n s of A. B. Theory of R e s i d u a l s . In what f o l l o w s , some a s p e c t s of the a r i t h m e t i c of submodules w i l l be d e v e l o p e d . T h i s may be done i n a g e n e r a l l a t t i c e s e t t i n g ( f o r example, the m u l t i p l i c a t i v e l a t -t i c e s of, L e s i e u r and C r o i s o t [9])# but f o r the purposes of t h i s t h e s i s modules are s u f f i c i e n t l y g e n e r a l . A l l of the p r o o f s of t h i s s e c t i o n may be used i n the l a t t i c e f o r m u l a t i o n w i t h o u t s i g -n i f i c a n t m o d i f i c a t i o n . 5 Throughout t h i s s e c t i o n S w i l l denote a g e n e r a l r i n g w i t h i d e a l s A, B, e . . and M w i l l be a l e f t R-module w i t h submodules D e f i n i t i o n 1; Given two submodules L and N of M, the s e t of i d e a l s A of R such t h a t AN cz L has a maximal element c a l l e d the l e f t r e s i d u a l of L by_ N and i s denoted by L—»N. I f AN c L and BN c L then c l e a r l y (A + B)N C, L so t h a t L—*N i s w e l l d e f i n e d i f the s e t of i d e a l s has a maximal element. Let A j £ Ag c ••• be an a s c e n d i n g c h a i n of i d e a l s such t h a t A^N c L f o r i = h, 2, ••• . Then, and Zorn's lemma g i v e s the r e s u l t . D e f i n i t i o n 2: Given a submodule L of M and an i d e a l A of R, the set of submodules N such t h a t AN c L has a maximal element c a l l e d the r i g h t r e s i d u a l of L b_y A and i s denoted L-t—A. The submodule L-s—A i s w e l l d e f i n e d by an argument s i m i l a r t o t h a t f o r L—*N. At t h i s stage i t i s u s e f u l t o note some si m p l e p r o p e r t i e s of r e s i d u a l s which w i l l be used f r e q u e n t l y . i n the s e q u e l . ... (a) A(L-s-A) c L, (L—»N)N CZ. L. (b) A 1 eg A 2 i m p l i e s Lt—k^ c Lt—k^; (c) - L, c L_ i m p l i e s L.-t—A cz L„-s—A and L. —»N cz L 5—»N. 6 (4) L-s-AB = (L-^A)-j-B. I f N c L-s-AB then ABN c L so t h a t BN a L-s-A and N c (L-s-A)-*-B. C o n v e r s e l y , i f N c (Lf-A)-s-B then BN c L-s-A so ABN c L. (e) W ( N 1 • M 2) = ( L - f N 1 ) n ( L - f N 2 ) ; ( L x + L 2 ) t - A 2 ( L 1 + - A ) " n ( L 2 t - A ) ; L * - ( A 1 • A 2 ) = (L<-A 1) D (L+-A 2).-' For many purposes a module over a g e n e r a l r i n g i s t o o broad a concept and the 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 u s e f u l r e s -t r i c t i o n : (RCC) A l e f t R-module M i s s a i d to have the r e s i d u a l c h a i n  c o n d i t i o n i f M has the a s c e n d i n g c h a i n c o n d i t i o n on r i g h t r e s i d u a l s and R has the a s c e n d i n g c h a i n c o n d i t i o n on l e f t r e s i d u a l s . By the u s u a l argument, (RCC) i s e q u i v a l e n t t o s a y i n g t h a t s e t s of l e f t o r r i g h t r e s i d u a l s have maximal elements. In what f o l l o w s t h i s c o n d i t i o n w i l l be r e f e r r e d t o by n a module M has (RCC)" o r a s i m i l a r p h r a s e . D e f i n i t i o n 3: An i d e a l P of R i s c a l l e d prime i f AB c P i m p l i e s t h a t e i t h e r A c P or B cz P. An e q u i v a l e n t f o r m u l a t i o n u s i n g elements i s : P i s prime i f aRb cz P i m p l i e s a £ P or b £ P ( [ 1 0 ] , Theorem l ) . D e f i n i t i o n hi An i d e a l A of R i s c a l l e d a p r o p e r l e f t r e s i d u a l of a submodule L of M i f A = L—*N f o r some N ^ L. 7 Lemma 5 : An i d e a l P which i s maximal i n the set of proper l e f t r e s i d u a l s of a submodule L i s prime. Proof: Let P = L-*N, N £ L. Assume AB c P with B £ P. We have ABN c L and .BN L by the maximality of P. Hence, A c L—rBN and L—s-_BN i s a proper l e f t r e s i d u a l which contains L—*N. Thus, L—s-BN = L—s-N and so A cz P. 3 The f o l l o w i n g i s obvious but i s worth n o t i n g . Lemma 6: If M has (RCC) then every submodule has a maximal proper l e f t r e s i d u a l . The next lemma gives a u s e f u l method f o r d e c i d i n g i f an i d e a l i s a proper l e f t r e s i d u a l . Lemma 7' An i d e a l A i s a proper l e f t r e s i d u a l of L i f , and only i f , L—? (L-s-A) = A where L-s-A •=> L. Proof: The c o n d i t i o n c l e a r l y i m p l i e s that A i s a proper l e f t r e s i d u a l . Conversely, l e t A = L—s-N, H £ L. Then, L-t—(L—s-N) p N s i n c e (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 since A c L-? (L-s-A) <- L—s-N = A. J | Theorem 8: I f M has (RCC) then to any proper, submodule L there corresponds a f i n i t e number of primes P^ such that P,P~ P M c L where P. i s a maximal proper l e f t r e s i d u a l — 1—2 —n — —1 of L-s-P,P0 ... P, 1 . —1—2 — i - i Proof: Let P^ be a maximum proper l e f t r e s i d u a l of L and l e t P, = L—s-N,. Let L-s—P. = K. and then K. D L by Lemma 7 and 8 P^ a L — 2 L—*h, I f + M we may consider a maximum proper l e f t r e s i d u a l °* a n d -2 = K l " ~ * Y 2 2 L"*'Y2. 2 L - r M ' Let Kg = K^f—Pg * L+-P^Pg. The process may be continued induc-t i v e l y and the process terminates a f t e r f i n i t e l y many steps by the (RCC). We have K R - M - L t-P^ Pg ••• J? n and the r e s u l t f o l l o w s * 9 Lemma 9: I f M has (RCC), L-t-A p L i f , and only i f # A i s con-t a i n e d i n . a maximal proper l e f t r e s i d u a l of L, Proof: I f L-t—A r» L then L—t(L-t-A) i s a proper l e f t r e s i d u a l of L and A tz\ L— t (Lt—A). Conversely, i f P i s a proper maximal l e f t r e s i d u a l of L with A c P = L—tN then L+-A 3 Lt-P • L t - ( L — » H ) 2 N so that L-t—A z> L since N ^ L. 1 Lemma 10: I f P i s a maximal proper l e f t r e s i d u a l of L and i f B gt P then P i s a proper l e f t r e s i d u a l of L+-B. Proof: Let C = (L - t-B)-•[ (L-t-B)-t-P] . Since P[(L+-B)+-P] c ( L - J - B ) , P C C. Conversely, L+-BC = (L-t-B)-t-C -(Lt-B)t-P since c l e a r l y (L-s-B)-t-C s (Lt-B)t-P by the f a c t that P c C but C[ (L+-B)i-P], s (Lt-B) by d e f i n i t i o n of C so (L-t-B)-t-P cz (Lt-B)t-C. Then, (L+-B)t-P = L+-BP p L-t—P. Thus BC cz L — t ( L - t — BC) cz L—t(L-t-P) - P. But, P i s prime and B ^ P so C c P. This gives the f i r s t c o n d i t i o n of Lemma 1. The 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 (Lt-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 c o n t r a d i c t i o n . J | 9 Theorem 11; I f M has (RCC) and L ^ M then L has o n l y f i n i t e l y many maximal p r o p e r l e f t r e s i d u a l s . P r o o f : Let P be a maximal p r o p e r l e f t r e s i d u a l of L. By Theorem 8 t h e r e i s a f i n i t e c o l l e c t i o n of primes P,, P„, •••• P — i —<i —n such t h a t P = L—*N 2 L—*M 2 P-^  Zn • S i n c e P i s maximal i t i s prime and so f o r some i , P^ c P. Assume P^ ^ P f o r j = 1, 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 of L i — P . ••• P. . = K, S i n c e P. i s a maximal p r o p e r l e f t r e s i d u a l 1 i - i i - i — i of K._ l f P c P i m p l i e s P = P.. J | Theorem 11 i m p l i e s t h a t the primes of Theorem 8 i n c l u d e a l l the maximal p r o p e r l e f t r e s i d u a l s of L. In l a t e r p a r t s of t h i s t h e s i s we w i l l need a c l a s s of p r o p e r l e f t r e s i d u a l s which are prime i d e a l s and the c l a s s i s l a r g e r than the c l a s s of maximal p r o p e r l e f t r e s i d u a l s . The f o l l o w i n g d e f i n i -t i o n g i v e s such a c l a s s and i s very u s e f u l i n the c h a r a c t e r i z a t i o n of t e r t i a r y i d e a l s i n P a r t I I . D e f i n i t i o n 12: 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 of a submodule L i f t h e r e i s a submodule N <£ L w i t h P = L—»N and i f K £ N and K ^ L then L-*K = L—tN. E s s e n t i a l r e s i d u a l s a r e , by d e f i n i t i o n , p r o p e r l e f t r e s i d u a l s . Theorem 13: Every maximal p r o p e r l e f t r e s i d u a l of a submodule L i s an e s s e n t i a l r e s i d u a l of L. 1 0 P r o o f : I f P i s a m a x i m a l p r o p e r l e f t r e s i d u a l o f L , P = L—s-N w h e r e N £ L . I f K £ L a n d K c N t h e n L - + K D K — S - N b u t L — t K = K — * N b y t h e m a x i m a l i t y o f P . J | T h e o r e m 1 4 : E v e r y e s s e n t i a l r e s i d u a l i s p r i m e . P r o o f : L e t P b e a n e s s e n t i a l r e s i d u a l p f L , P=> L—s -N w i t h N ^ L , a n d a s s u m e t h a t BC c P w i t h C ^ P . T h e n , BCN c L a n d CN ^ L . S i n c e CN c N a n d L - * N i s e s s e n t i a l , L-—s-CN = L—s-N =P a n d B c L — * C N = P . T h e o r e m 1 5 : I f M h a s ( R C C ) e v e r y s u b m o d u l e L o f M h a s o n l y f i n i t e l y m a n y p r i m e p r o p e r l e f t r e s i d u a l s . P r o o f : L e t a b e a n i n f i n i t e s e t o f p r i m e p r o p e r l e f t r e s i d u a l s o f L . B y t h e ( R C C ) o h a s a m a x i m a l e l e m e n t _ B ^ . L e t N ^ = L-s—JB^, 3 L b y L e m m a 7« By L e m m a 1 0 , e v e r y e l e m e n t o f a i s a p r i m e p r o p e r l e f t r e s i d u a l o f N ^ . L e t b e a m a x i m a l e l e m e n t o f a \ i j 1 ] a n d l e t N 2 = L t - B ^ = N ^ * - J 2 3 N j , T h u s 3 D N . C o n t i n u i n g t h e p r o c e s s l e a d s t o a n i n f i n i t e c h a i n o f s u b m o d u l e s w h i c h a r e r i g h t r e s i d u a l s — c o n t r a d i c t i n g ( R C C ) . J | C o r o l l a r y 1 6 : I f M h a s ( R C C ) e v e r y s u b m o d u l e L ^ M h a s o n l y f i n i t e l y m a n y e s s e n t i a l r e s i d u a l s . P r o o f : T h e o r e m 14 a n d T h e o r e m 15 g i v e s t h e r e s u l t . JH T h e u s e f u l n e s s o f t h e t h e o r y o f r e s i d u a l s a s d e v e l o p e d h e r e i s m o r e a p p a r e n t i f i t i s n o t i c e d t h a t a l l t h e p r o o f s w o u l d f o l -l o w i f , i n s t e a d o f s u b m o d u l e s , t w o - s i d e d i d e a l s w e r e u s e d 11 t h r o u g h o u t . The c o n d i t i o n (RCC) i s seen t o be u s e f u l i f we note the f o l l o w i n g examples when (RCC) h o l d s . (1) M and R s a t i s f y the a s c e n d i n g c h a i n c o n d i t i o n (ACC) (2) M and R s a t i s f y the d e s c e n d i n g c h a i n c o n d i t i o n (DCC) (3) M s a t i s f i e s both (ACC) and (DCC) (4) R s a t i s f i e s both (ACC) and (DCC). 12 PART II G e n e r a l i z a t i o n s of the Decomposition Theorem and the F i r s t Uniqueness Theorem A Primary i d e a l s and primary submodules D e f i n i t i o n 1; A subset H of a r i n g R i s c a l l e d an m-system i f e i t h e r ( i ) M = 0 or ( i i ) a, b £ M imply that there e x i s t s x £ R such that axb £ M. M u l t i p l i c a t i v e l y c l o s e d sets and complements of prime i d e a l s are c l e a r l y ra-systems. D e f i n i t i o n 2: The r a d i c a l r(A) of an i d e a l A i s the set of elements r £ R with the p r o p e r t y that every m-system con-t a i n i n g r meets A. Theorem 3: The r a d i c a l r(A) of an i d e a l A i n R i s the i n t e r -s e c t i o n of a l l prime i d e a l s which are minimal i n the set of primes c o n t a i n i n g A ( [ 1 0 ] , Theorem 2 ) . D e f i n i t i o n Ui An element a £ R i s s a i d to be r i g h t prime (rp) to an i d e a l A i f aRx c A i m p l i e s that x £ A. An i d e a l B i s r i g h t prime (rp) to A i f i t contains an element which i s rp to A. The negation i s c a l l e d not r i g h t prime ( n r p ) . Def i n i t ion 5 : An i d e a l _Q i s s a i d to be r i g h t primary (or j u s t primary) i f aRb c and a t r(Q) together imply b £ Q. Lemma 6: If R i s commutative, the r i g h t primary i d e a l s are the usual primary i d e a l s . 13 P r o o f : T h i s f o l l o w s 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 p r i m a r y i f , and o n l y i f , every element not i n T(Q) i s rp to JJ; and from the f a c t t h a t the r a d i c a l of an i d e a l i n a commutative r i n g i s the i n t e r s e c t i o n of a l l the mini m a l prime i d e a l s con-t a i n i n g the i d e a l . H Theorem 7: I f R has e i t h e r the a s c e n d i n g o r the des c e n d i n g c h a i n c o n d i t i o n then the r a d i c a l of a r i g h t p r i m a r y i d e a l i s p r i m e . T h i s theorem. Lemma 8 and Theorem 11 w i l l not be pr o v e d i n the p r e s e n t form. P r o o f s may be found i n [ 1 1 ] . These r e s u l t s w i l l be r e s t a t e d , and prove d , i n a more g e n e r a l sense l a t e r i n t h i s t h e s i s . Lemma 8: I f R has e i t h e r the a s c e n d i n g o r the d e s c e n d i n g c h a i n c o n d i t i o n f o r i d e a l s then every i d e a l has a mi n i m a l prime d i v i s o r which i s nrp t o i t . D e f i n i t i o n 9: I f an i d e a l A i s an i n t e r s e c t i o n of f i n i t e l y many r i g h t p r i m a r y i d e a l s (i) A - 5 r n ••• n Qk such 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 p r i m a r y 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 + i n """ n -S.k i s c o n t a i n e d i n i = 1, k. The p r i m a r y r e p r e s e n t a t i o n ( l ) i s c a l l e d s h o r t i f none of the i d e a l s o b t a i n e d by t a k i n g the i n t e r s e c t i o n of two or more of the i d e a l s Q^, Qj, are p r i m a r y . 14 The f o l l o w i n g theorem i s a summary of some of the more imp o r t a n t r e s u l t s which f o l l o w from D e f i n i t i o n 5 (of p r i m a r y i d e a l s ) . The theorem i s due t o Murdoch [ 1 1 ] . T h e o r e m 10: Let R be a r i n g w i t h the a s c e n d i n g c h a i n c o n d i -t i o n f o r i d e a l s . Let A be an i d e a l which has a p r i m a r y r e p r e s e n t a t i o n . Then A has 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 . I f A has two s h o r t r e p r e s e n t a t i o n s A = g1 n • • • n £ k - 9[ n • • • n t h e n k = t and the s e t s of prime i d e a l s (r(Q^) : i= 1 ,•••,k} a n d [ r ( Q j ) : j = 1, {,} c o i n c i d e i n some o r d e r . Example 1 of P a r t IV shows t h a t the a s c e n d i n g c h a i n con-d i t i o n i s not s u f f i c i e n t to guarantee the e x i s t e n c e of p r i m a r y r e p r e s e n t a t i o n s . The i d e a s o u t l i n e d above can be extended t o the m u l t i p l i c a -t i v e l a t t i c e s of [ 9 ] . In what f o l l o w s the g e n e r a l i z a t i o n w i l l be d e s c r i b e d i n the language of modules but the f o l l o w i n g remarks i n t h e language of t w o - s i d e d i d e a l s are u s e f u l f o r m o t i v a t i o n . In a r i n g R, the i d e a l A—s-R i s an i d e a l c o n t a i n i n g A. Lemma 11 : r ( A ) = r (A—s-R). P r o o f : Let x £ r ( A ) . Then i f P i s a prime c o n t a i n i n g A, x £ P. I f P i s a prime c o n t a i n i n g r (A—s-R) then A c P so x £ P and r ( A ) c r(A^-s-R). Let P be a m i n i m a l prime i n the set of primes c o n t a i n i n g A such t h a t A—*R ^ P. That i s , l e t t h e r e e x i s t x £ A—s-R such t h a t x 4 P. x £ A—*R i m p l i e s xR c A so 15 xRx cr A. Hence xRx c P, x t P which c o n t r a d i c t s the f a c t that P i s prime. 9 D e f i n i t i o n 12: I f M i s a l e f t R-module and L i s a submodule of M, d e f i n e the r a d i c a l of L as r(L—tM) where r ( L — t M ) i s the r a d i c a l of the i d e a l L — t M as i n D e f i n i t i o n 2. The r a d i c a l of L i s denoted by r ( L ) . Theorem 13: I f M has (RCC) and L i s a submodule of M, r ( L ) i s the maximal element of the set of i d e a l s A such that A m cr L — t M f o r some i n t e g e r m. Proof: I f L = M, M—tM = R and r ( L ) = R. If L * M, there e x i s t s by Theorem 8, Part I, a f i n i t e number of primes P^, i = 1, - «•, k, such that Pj. ro L — » M and P^ f 2 ... JPfc er L — t M . The minimal primes i n the set (_P^  : i = 1, k} i n c l u d e the minimal primes c o n t a i n i n g L—tM s i n c e P^ ... JP^ , cr L — t M c P^. I f k A r a c L—s-M c P then A c P, and A c (~) P. . Conversely, k 1 i = l Lemma 14: If L , , L are submodules then a 1* n r ( L n ••• 0 L ) = r ( L . ) n ••• n r ( L ) . l n i n Proof: T h i s f o l l o w s s i n c e ( L X n ••• n L n ) ~ » M = ( L 1 - t M ) n n ( L N — » M ) . a D e f i n i t i o n 15: If M i s a l e f t R-module, a submodule Q of M i s c a l l e d primary 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 c o n d i t i o n s are s a t i s f i e d . 16 ( i ) AL c Q, L £ Q i m p l i e s A cz r ( Q ) . ( i i ) A ^ r(Q) i m p l i e s Q-s-A = Q. In view of Theorem 13, t h i s d e f i n i t i o n may be reworded i n the case where M has (RCC) t o ( i i i ) AL c Q, L Q then t h e r e e x i s t s a p o s i t i v e i n t e g e r k such t h a t A kM £ Q. Theorem 16: I f M has (RCC) then Q M i s p r i m a r y i f , and o n l y i f , Q has e x a c t l y one p r o p e r prime l e f t r e s i d u a l and t h i s prime r e s i d u a l i s m i n i m a l i n the s e t of i d e a l s c o n t a i n i n g Q-s-M. T h i s prime i s r ( Q ) . P r o o f : Let P_ be a p r o p e r prime l e f t r e s i d u a l o f a p r i m a r y submodule Q ^ M. By Lemma 9 of P a r t I , Q+-P 3 Q so P c r ( Q ) . Hence P = r(Q) and so _P i s unique and s i n c e every prime con-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 . C o n v e r s e l y , i f P i s a unique m i n i m a l prime c o n t a i n i n g Q—s-M then P = r ( Q ) . I f P i s the o n l y p r o p e r prime l e f t r e s i d u a l , i f A P then Q+-A = Q by Lemma 9, P a r t I . So, Q i s p r i m a r y . J| A p r i m a r y submodule w i t h r a d i c a l P w i l l be c a l l e d P -primary. The i n t e r s e c t i o n of f i n i t e l y many P-primary submodules i s again P-primary i f P i s p r i m e . T h i s i s t r u e s i n c e i f A £ P, Q^-s—A = Q 1, Qg+^ -A = Q E i f Q 1 and Q 2 are P-primary. Then, (Q1 0 0-2^ *""- = Q l 0 Q2 s o Q l 0 Q2 i s P - P r i r a a r v s i n c e r ( Q x Q Q 2) = r ( Q 1 ) 0 r ( Q 2 ) = P. 17 D e f i n i t i o n 1 7 : I f a s u b m o d u l e L i s e x p r e s s i b l e a s a n i n t e r s e c -t i o n o f f i n i t e l y m a n y p r i m a r y s u b m o d u l e s (2) _ L = Q 1 n • • • n Q k s u c h a n e x p r e s s i o n 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 . T h e e x p r e s s i o n (2) i s c a l l e d i r r e d u n d a n t - i f n o i n t e r -s e c t i o n Q 1 n • • • H D PI ••• n Q k i s c o n t a i n e d i n Q .^ T h e e x p r e s s i o n (2) i s c a l l e d s h o r t i f n o n e o f t h e s u b m o d u l e s o b t a i n e d b y t a k i n g t h e i n t e r s e c t i o n o f t w o o r m o r e o f t h e s u b m o d u l e s Q ,^ Qj , i s p r i m a r y . L e m m a 1 8 : I f M h a s (RCC) a n d (2) i s a n i r r e d u 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 t h e n L-t—A = L i f , a n d o n l y i f , A gt P ^ f o r i = 1, k w h e r e P^ = r(Q^). P r o o f I f L-t—A cj: L , t h e r e e x i s t s N g£ L s u c h t h a t A H c L cr Q i 0 B u t , N £ L = Q 1 f) • • • D Qk s o L ± Q.. f o r s o m e i . A N c Q., N £ Q. s o A c r(Q.) = P . s i n c e Q . i s p r i m a r y . ( i i ) I f A c P , f o r s o m e i , a s s u m e A c P ^ . T h e n , b y T h e o r e m 13, t h e r e i s a n i n t e g e r m ( a s s u m e d m i n i m a l ) s u c h t h a t A m M cr Q x . We w i l l s h o w t h a t L-s—A £ L . N o w , Q 3 A r a M ro A m (Q2 0 n Qk) a n d b y t h e m i n i m a l i t y o f m e i t h e r Q2 n • • • n Qk c L-s—A ( i f m = 1) o r A m " 1 ( Q 2 n fl Q fc) cr L-t—A ( i f m > l ) . B u t t h e m i n i m a l i t y o f m i m p l i e s t h a t A r a ~ 1 ( Q 2 n • • • n Qk) £ L . H L e m m a 1 9 I n a m o d u l e w i t h ( R C C ) , a n i r r e d u n d a n t i n t e r s e c t i o n o f p r i m a r y s u b m o d u l e s n o t a l l o f w h i c h h a v e t h e s a m e r a d i c a l i s n o t p r i m a r y . 18 P r o o f : I t i s s u f f i c i e n t t o show the r e s u l t f o r two p r i m a r y submodules Q^, Q 2 which are not r e l a t e d by i n c l u s i o n . Let r ( Q x ) = - P 1 , r ^ 2 ^ = P 2 ' Q = Q l n Q 2 * A s s u m e Q i s P - p r i m a r y , then P c P j (1 P 2» The i d e a l P i s a p r o p e r l e f t r e s i d u a l of Q by Theorem 16, so Q-t-P => Q. I f Q-s—A = Q then Lemma 18 i m p l i e s t h a t A £ ? 1 and A ± V.2' T h e r e f o r e , Q - J-j^ 3 Q and Q - i - . E 2 => Q. Hence, f o r some L^, P^L 1 cz Q, ^ Q so P j c P 8 r(Q) s i n c e Q i s p r i m a r y . S i m i l a r l y P_2 cz P and P = P 1 = P 2« J | I t i s now p o s s i b l e t o prove the f i r s t uniqueness theorem d i r e c t l y , but i t i s a consequence of Theorem 21 of P a r t I I I and a s p e c i a l case of Theorem 30 of t h i s p a r t so i t w i l l j u s t be s t a t e d h e r e . Theorem 20: I f M i s a module w i t h (RCC) and i f L has a p r i m a r y r e p r e s e n t a t i o n , then any two 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 s of L have the same l e n g t h and the r a d i c a l s of the p r i m a r y com-ponents of the two i n t e r s e c t i o n s c o i n c i d e i n some o r d e r . B. T e r t i a r y submodules. The d e f i n i t i o n of t e r t i a r y i d e a l s and submodules i s due t o L e s i e u r and C r o i s o t and they have developed these i d e a s i n terms o f elements and i n the language of m u l t i p l i -c a t i v e l a t t i c e s ( [ 5 ] , [ 6 ] , [ 7 ] and [ 9 ] ) . Most of what i s quoted below i s i n terms of the l a t t i c e f o r m u l a t i o n s p e c i a l i z e d t o l e f t modules over a r b i t r a r y r i n g s . The d i s c u s s i o n i n terms of elements i s a l r e a d y a v a i l a b l e i n E n g l i s h [4]« 19 D e f i n i t i o n 21: A submodule Q of an R-raodule M i s s a i d t o be t e r t i a r y i f Q-e—A 3 Q and (Q-s-A) n L = Q i m p l y L = Q. Every n - i r r e d u c i b l e module i s c l e a r l y t e r t i a r y . Let M = M - Q and X denote the image of a submodule X of M under the n a t u r a l homomorphism of M i n t o M. A submodule Q i s t e r t i a r y i f Qi—A £ 0 i m p l i e s t h a t Q-e—A i s l a r g e ( i . e . , has non-zero i n t e r -s e c t i o n w i t h e v e r y non-zero submodule of M). D e f i n i t i o n 22: The t e r t i a r y r a d i c a l of a submodule L of M, denoted by ?v.(L), i s the maximum i d e a l of R of the form A where (L-s—A) 0 N c L i m p l i e s N c L. In P a r t IV an e q u i v a l e n t d e f i n i t i o n of X(h) i s g i v e n i n terms of the elements of the module and r i n g . With t h a t d e f i n i t i o n i t i s 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 , and so we w i l l not attempt t o show t h a 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 o n l y the a r i t h m e t i c of submodules and i d e a l s . However, the next theorem does t h i s In the case t h a t M has (RCC). Theorem 23: I f M has (RCC), L a submodule of M, then K(L) i s the i n t e r s e c t i o n of the e s s e n t i a l r e s i d u a l s of L. P r o o f : I t f o l l o w s i m m e d i a t e l y from D e f i n i t i o n 22 t h a t \(L) i s t he maximum i d e a l of the form A where (L-s—A) n N = L i m p l i e s N = L. ( i ) Let P be an e s s e n t i a l r e s i d u a l of L. There i s N z> L such 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. L e t A c A.(L) and s a t i s f y the c o n d i t i o n s t a t e d above. 20 (3) A g L-t(L-t-A) c L - t [ ( L t - A ) 0 H] L c N s o (L-s—A) fl H 3 L b y t h e c o n d i t i o n o n A . H e n c e , L c (L-t—A ) n N c N s o L - * [ ( L t - A ) f| N] = L—tN = P . C o m b i n i n g -t h i s w i t h (3)# A c P a n d s o X.(L) ez P f o r e v e r y e s s e n t i a l r e s i d u a l P o f L a n d s o K(L) i s c o n t a i n e d i n t h e i r i n t e r s e c t i o n . k ( i i ) L e t JL = P i w h e r e P 1 # P f c a r e a l l t h e e s s e n t i a l i = l r e s i d u a l s o f L. A s s u m e (L-t—J.) n N = L a n d L c N. T h e n , L—tN 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 f L ez K ez N i m p l i e s L—tK = L—tN t h e n P = L—tN i s a n e s s e n t i a l r e s i d u a l o f L a n d J_ e P . T h e n , L t - X 2 L-s—P 3 N w h e n c e N = ( L t — I) Cl N = L c o n t r a d i c t i n g t h e f a c t t h a t N => L. I f L c K c N d o e s n o t i m p l y L - ^ K = L—tN t h e n i t i s p o s s i b l e t o c h o o s e N^ c N s u c h t h a t N^ r> L a n d L—tN^ 3 L — t N . I f we h a v e N r s u c h t h a t L cz K cz N^ d o e s n o t i m p l y L—tK = L—tN^ c h o o s e N . r N s u c h t h a t N . 3 L a n d L-*N ^. 3 L—tN . By t h e ( R C C ) r+1 r r+1 r+1 r ' t h e a s c e n d i n g c h a i n o f r i g h t r e s i d u a l s s t a b i l i z e s s o t h a t t h e r e i s a n N 1 cz N, N* 3 ' L s u c h t h a t P = L — t N 1 i s a n e s s e n t i a l r e s i d u a l o f L. A g a i n I_ c P , L t - l 3 L-t -P 3 N* a n d a l s o N» - ( L - t - l ) f) N* ez ( L - t - l ) 0 N = L c o n t r a d i c t i n g N* 3 L. T h u s A = _I s a t i s f i e s t h e k c o n d i t i o n a n d A.(L) = f \ P . • J| i = l 1 I t i s n o w p o s s i b l e t o d e v e l o p t h e m o r e i m p o r t a n t p r o p e r t i e s o f t e r t i a r y s u b m o d u l e s w h i c h w i l l l e a d t o a d e c o m p o s i t i o n t h e o r e m a n d t h e f i r s t u n i q u e n e s s t h e o r e m w i t h o n l y ( R C C ) a s a n a s s u m p -t i o n . L e m m a 2k; I f L^, L^ a r e s u b m o d u l e s o f M t h e n n •»• n L k ) 3 K C L ^ n • • • n * . ( L k ) . 21 j P r o o f : L e t k = 2 a n d l e t \ ( L 1 ) fl X . ( L 2 ) = A . A s s u m e t h e r e e x i s t s N s u c h t h a t [ (L^^ n L 2 ) - s - A ] n N cz L 1 n L g . Now ( L 1 - s - A ) n ( L 2 t - A ) = ( L 1 n L 2 ) i - A s o ( l ^ i - A ) fl (L 2+-A) n H c L l n L2 - L l ' a n d S O ^L2"*~-^ ^ N — L l - L i * ~ ^ s i n c e A c X ( L 1 ) « T h u s ( L 2 - « - A ) n N = ( L j ^ - s - A ) fl (L2*~-^ n N £ L l 0 L2 S L 2 ; a n d A c ? v . ( L 2 ) i m p l i e s N Q l>2 cz h^r-k. H e n c e , N = ( L 2 - s — A ) n N c L1 a n d i t f o l l o w s t h a t N c L ^ n L 2 « T h i s g i v e s A cz \(L^ fl L 2 ) . I n d u c t i o n g i v e s t h e l e m m a . J | T h e r e v e r s e i n e q u a l i t y d o e s n o t h o l d e v e n w i t h ( R C C ) ([5], e x a m p l e 7.1). T h e o r e m 25 : ( l ) A s u b m o d u l e Q o f M i s t e r t i a r y i f , a n d o n l y i f , A N cz Q , N £ Q i m p l y A c \{Q) o r , e q u i v a l e n t l y , A £ \ ( Q ) i m p l i e s Q-s—A = Q . ( i i ) I f M h a s ( R C C ) a s u b m o d u l e Q i s t e r t i a r y i f , a n d o n l y i f , Q h a s e x a c t l y o n e e s s e n t i a l r e s i d u a l P a n d A . ( Q ) - P . P r o o f : ( i ) I f Q i s t e r t i a r y a n d A N cz Q, N £ Q t h e n Q - s - A 3 Q s o b y D e f i n i t i o n 21, ( Q - s - A ) n K = Q i m p l i e s K = Q a n d D e f i n i t i o n 22 g i v e s A c k ( Q ) . C o n v e r s e l y , i f A N cz Q , N £ Q i m p l y A c ~k(Q) t h e n Q-s—A zo Q a n d ( Q - s - A ) n L = Q i m p l y A c X.(Q) s o L = Q a n d Q i s t e r t i a r y . ( i i ) I f Q i s t e r t i a r y a n d P i s a n e s s e n t i a l r e s i d u a l o f Q , Q t - P 3 Q s o P c A . ( Q ) . B u t , T h e o r e m 23 g i v e s \(Q) a P s o t h a t P i s u n i q u e a n d e q u a l t o ?v , (Q) . C o n v e r s e l y , i f P i s t h e u n i q u e e s s e n t i a l r e s i d u a l o f Q , A . ( Q ) = P . S i n c e Q h a s m a x i m a l p r o p e r l e f t r e s i d u a l s , P m u s t b e m a x i m a l . H e n c e A £ P i m p l i e s Q-s—A = Q s o Q i s t e r t i a r y b y ( i ) . S 22 As an immediate c o r o l l a r y t o t h i s theorem, ( R C C ) i m p l i e s t h a t the t e r t i a r y 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 . I f Q i s t e r -t i a r y and X ( Q ) = P we w i l l c a l l Q P - t e r t i a r v . Lemma 26: I f and Q 2 are two P - t e r t i a r y submodules, so i s ^1 ^ ^2* S i m i l a r l y f o r any f i n i t e number o f P - t e r t i a r y sub-modules. P r o o f : X ( Q X n Q 2 ) = X . ( Q X ) fl ? v.(Q 2) = P. I f AN cr 0^ n Q 2 then N <^  Q 1 n Q 2 i m p l i e s e i t h e r N £ Q1 or N £ Q 2 » E i t h e r case i m p l i e s A cr P. T h i s i s t r u e , i n p a r t i c u l a r , f o r A = X . ( Q 1 n Q 2 ) . I D e f i n i t i o n 27: I f a submodule L can be e x p r e s s e d as an i n t e r s e c -t i o n of f i n i t e l y many t e r t i a r y submodules (4) L = Q 1 n • • • n Q K then (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 of L. I f the i n t e r s e c t i o n i s i r r e d u n d a n t and the i d e a l s A . ( Q ^ ) , X ( Q ^ ) are d i s t i n c t then the t e r t i a r y r e p r e s e n t a t i o n (4) i s c a l l e d reduced. The d i s t i n c t i o n between the d e f i n i t i o n of "reduced 1* and the d e f i -n i t i o n of " s h o r t 1 * i n D e f i n i t i o n 17 i s i m p o r t a n t s i n c e i t w i l l a l l o w us to note t h a t the f i r s t d e c o m p o s i t i o n theorem f o r p r i m a r y r e p r e s e n t a t i o n s i s a s p e c i a l case of t h a t f o r t e r t i a r y decomposi-t i o n s . Theorem 28: I f M has ( H C C ) then every submodule has a reduced t e r t i a r y r e p r e s e n t a t i o n . 23 I f the a s c e n d i n g c h a i n c o n d i t i o n f o r submodules i s assumed then the theorem f o l l o w s i m m e d i a t e l y from Lemma 26 s i n c e e v e r y i r r e d u -c i b l e submodule i s t e r t i a r y . For the more g e n e r a l case o f (RCC) the r e s u l t f o l l o w s from Lemma 37 below. The p r o o f of Lemma 37 i n v o l v e s p r i m a l submodules which are d e f i n e d and d i s c u s s e d i n s e c t i o n C of t h i s p a r t . Note t h a t s i n c e Lemma 26 d i d not r e q u i r e (RCC), the e x i s t e n c e of a f i n i t e t e r t i a r y r e p r e s e n t a t i o n i m p l i e s the e x i s t e n c e of a reduced r e p r e s e n t a t i o n . Lemma 29: I f and Q 2 are t e r t i a r y submodules, of a module M w i t h (RCC), w i t h t e r t i a r y r a d i c a l s P^ and P 2 r e s p e c t i v e l y , i f px £ L2 t h e n L = Q l n N l = Q 2 0 N 2 i m P l i e s L = N 1 n N 2 . P r o o f : We have Qg"8"-2-i = ®2 s * n c e n o t , P^ i s c o n t a i n e d i n a maximal p r o p e r l e f t r e s i d u a l of Q 2 which must be P 2» Hence, L f - P j - , - ( Q i t - P i ) 0 ( N J L - S - P J ^ ) = Q 2 fl ( N 2 - s - P 1 ) . I n t e r s e c t the l a s t two e x p r e s s i o n s w i t h N^ fl N 2 n o t i n g t h a t N^-s—P^ p N^ and H2'l"-1 - N 2 " T h i s g i v e s ^ l " 8 - " - ^ n N l n N 2 = Q 2 0 N l 0 N 2 = L - Q l and, s i n c e Q 1 i s P ^ - t e r t i a r y , fl ^ £ s o L = Q X n N X n N 2 = N X n N 2 . 3 T h i s lemma i s the t o o l w i t h which the f i r s t uniqueness theorem f o r reduced t e r t i a r y r e p r e s e n t a t i o n s may be prov e d . Theorem 30: I f M has (RCC) and a submodule L has two reduced t e r t i a r y r e p r e s e n t a t i o n L = Q 1 D n Q K = Q j n • • • n Q m 24 then k = m and the s e t s of primes {P^: P^ = r ( Q ^ ) , 1 < i < k} and ]P« : P» = r ( Q J ) f 1 < j < ra} c o i n c i d e i n some o r d e r . P r o o f : I t i s s u f f i c i e n t to show t h a t P^ = P_j f o r some j , 1 < j < nit Assume P^ i s d i f f e r e n t from a l l the P*. Then e i t h e r ( i ) P x £ P» o r P» £ P x ( i i ) P x £ P» or V2±lx . . . (•) Sx ± IL o r P m £ P.. By e i t h e r a l t e r n a t i v e of ( i ) and Lemma 29, L = Q * n ••• n n Q 2 n ••• n Q K . But L = Q » n • • • fl Q F F L n Q 2 n • • • n Q K = QL fl • • • n Q K and so e i t h e r a l t e r n a t i v e of ( i i ) p e r m i t s the use of Lemma 29 t o g i v e L = fl • • • n Q M 0 Q 2 0 ••• PI Q K - Given L = Q £ ft • " • fl Q M 0 Q 2 n fl Q K , r < m, e i t h e r a l t e r n a t i v e of ( r ) g i v e s L = Q * + 1 H ••• D Q M D Q 2 0 ••" D Q K » By i n d u c t i o n , L = Q 2 fl D Q K c o n t r a d i c t i n g the i r r e d u n d a n c y of a reduced t e r t i a r y r e p r e s e n t a t i o n . J | In what f o l l o w s , the u n i q u e l y d e t e r m i n e d primes a s s o c i a t e d w i t h a reduced t e r t i a r y d e c o m p o s i t i o n w i l l be c h a r a c t e r i z e d i n terms of e s s e n t i a l r e s i d u a l s . Lemma 31; Let L = n • • • fi Q F C be a reduced t e r t i a r y r e p r e s e n t a -t i o n of L i n a module w i t h ( R C C ) . Let P = L—»N be an e s s e n t i a l r e s i d u a l of L such t h a t L c K c N i m p l i e s L—*K = L—s-N = P. Then, t h e r e e x i s t s e x a c t l y one t e r t i a r y submodule such t h a t L = Q . n N. 25 P r o o f s We have H D L so ^ fl fl Qfc ft N = L. Let Q± PI h Q m D H = L where [Q^, Q m} i s m i n i m a l ( i . e . , i f Q. D HQ. ft N = L then r > m). S i n c e N 3 L, m > 1 and X l a r N» = Q 0 fl ••• fl Q M fl N 3 L. Thus L e H » e N and P = L-sN». Now, m • . —- — L—*N * - 0^—sN» s i n c e Q 1 3 L g i v e s L—s-N * cz 0^ —frN» but i f AN 1 c Q 1 # AN 1 c L by d e f i n i t i o n of N». But ? 1 = KiQ^ 3 P s i n c e ' Zx i s the unique e s s e n t i a l r e s i d u a l of Q^. But i f P^K cz and K c N* then P 1K £ L so ( I r t - P j ) n N» = ( Q l < _ P 1 ) Q N» £ Q1 s i n c e i s P^-t e r t i a r y . Hence, L cz (L«—P^) fl . N * cz N. T h e r e f o r e , P = L-tN = L - t C d - * - ^ ) n N*] 2 L-S-^L-J-PJ^) 3 P x and P - J ^ . I f m > 1 the same argument (changing o r d e r of terms i n the i n t e r s e c -t i o n ) g i v e s P = P 2 c o n t r a d i c t i n g the d e f i n i t i o n of a reduced t e r t i a r y r e p r e s e n t a t i o n . Hence fl N = L and r - ' Q 2 n ••••• n Q K n N 3 L . J | Theorem 32; Let L = fl • • • fl Q k be a reduced t e r t i a r y r e p r e s e n -t a t i o n of L i n a module w i t h (RCC). Then, the s e t of primes |P. - X ( Q . ) : 1 < i < k} i s e x a c t l y the set of e s s e n t i a l r e s i d u a l s of L. P r o o f : By Lemma 31 , every e s s e n t i a l r e s i d u a l i s i n c l u d e d i n the s et [ P ^ l i To show t h a t P^ i s an e s s e n t i a l r e s i d u a l of L, c o n s i d e r L 1 = Q 2 D " °» H Qj, and c o n s i d e r a l l l e f t r e s i d u a l s of L of the form L—s-N where L cz N c L». By (RCC), t h i s s e t has a maximal element P_ = L—s-K, L cr K cz L*. T h i s i s , by i t s c o n s t r u c -t i o n , an e s s e n t i a l r e s i d u a l of L. K fl Q 2 fl ••• Ct Qfc = K fl L* = K 3 L. But, by Lemma 31 , fl K = L f o r some i , i must be 1. T h e r e f o r e L = Q. fl K and £ = P,. By renumbering, t h i s method 26 shows that each of the primes P^r Pj, i s an e s s e n t i a l r e s i d u a l of L, J | This very u s e f u l theorem has many c o r o l l a r i e s , two of which are noted here* C o r o l l a r y 33 : Under the hypotheses of Theorem 32, - Zx n • • • n P k. Proof: X.(L) i s the i n t e r s e c t i o n of the e s s e n t i a l r e s i d u a l s of L. (Theorem 2 3 ) . j| C o r o l l a r y 34: Under the hypotheses of Theorem 32, P^# Pj , include the maximal proper l e f t r e s i d u a l s Of L, It now can be shown that i n a module with (RCC), r ( L ) g \(L) f o r any submodule L. By d e f i n i t i o n , r ( L ) i s the i n t e r s e c t i o n of the minimal primes c o n t a i n i n g L—tM. Except i n the t r i v i a l case where L = M these primes are proper i d e a l s and Theorem 8 of Part I and Theorem 13 show that these primes are proper l e f t r e s i d u a l s . The r e s u l t f o l l o w s i f i t can be shown that every e s s e n t i a l r e s i -dual contains one of the minimal primes. This i s a consequence of the (RCC). Let £ 1 55 L — t t ^ r> P 2 » L"~*N2 3 " " b e a d e s c e n d i n 3 chain of proper prime l e f t r e s i d u a l s of L which are contained i n an e s s e n t i a l r e s i d u a l P. C l e a r l y P^ «* L—t(L-t—P^) so that L+-P1 c L*~P 2 c a n d t h e ( R C C ) im p l i e s that the chain s t a b i -l i z e s and the set of primes has a minimal element. In a module M with (RCC) a primary submodule has e x a c t l y one prime proper l e f t r e s i d u a l (which i s minimum i n t h e ' s e t " o f ' l e f t 27 r e s i d u a l s c o n t a i n i n g L—tM). T h i s unique prime r e s i d u a l must be e s s e n t i a l s i n c e every submodule has f i n i t e l y many e s s e n t i a l r e s i d u a l s which are p r i m e . Hence a p r i m a r y submodule i s t e r -t i a r y and the two r a d i c a l s c o i n c i d e i n t h i s c a s e . I f L = fi • • • 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 then s i n c e r(Q^) f r ( Q j ) i f i ^ j i t i s a reduced t e r t i a r y r e p r e s e n t a t i o n . S i n c e the primes r(Q^) = X(Q^) are u n i q u e l y d e t e r m i n e d by Theo-rem 30 , we may conclude t h a t the i r r e d u n d a n t i n t e r s e c t i o n of two p r i m a r y submodules whose r a d i c a l s are d i s t i n c t i s not p r i m a r y (Lemma 1 ° ) and a l s o i s not t e r t i a r y . The t i t l e of t h i s p a r t i m p l i e s t h a t t e r t i a r y submodules are a g e n e r a l i z a t i o n of the c l a s s i c a l p r i m a r y submodules. That t h i s i s t r u e , at l e a s t f o r modules w i t h (RCC) i s shown by Theorem 33: I f M i s a l e f t R-module w i t h (RCC) and i f R i s commutative then a submodule L i s t e r t i a r y i f , and o n l y i f , i t i s p r i m a r y . P r o o f : L e s i e u r and C r o i s o t ([9], p. 74) have g i v e n a s u f f i c i e n t c o n d i t i o n t h a t every submodule which i s t e r t i a r y i s p r i m a r y . I t i s as f o l l o w s : ( i ) an i d e a l A i s c a l l e d M - p r i n c i p a l i f N c AL i m p l i e s t h e r e e x i s t s L» c L such t h a t N = AL», ( i i ) i f a l e f t R-module M has (RCC) and every i d e a l of R can be e x p r e s s e d as a sura of M - p r i n c i p a l i d e a l s , then e v e r y t e r t i a r y submodule i s p r i m a r y . We show t h a t e v e r y commutative r i n g R i s such t h a t the p r i n c i p a l i d e a l s are M - p r i n c i p a l f o r e v e r y module M over R. C l e a r l y e v e r y i d e a l i s the sum of p r i n c i p a l i d e a l s . 28 Let ( a ) L 3 N, a e R. Let L» = N-s-(a) = [m e M: (a)m c N}. C l e a r l y ( a ) L * c N. But, (a)m c N i f , and o n l y i f , am e N: (a)m c N i m p l i e s am e N, c o n v e r s e l y am e M i m p l i e s (Ra)m o N and ( i a ) m = i(am) e N f o r any i n t e g e r i ; the commutively g i v e s (a)m cz N. Thus i f (a)L» + N, t h e r e i s n e M such t h a t n t ( a ) L ' . But n e ( a ) L so n = ( r a * i a ) r a , m e L, which i m p l i e s rra • im £ L* — a c o n t r a d i c t i o n . 3 The s u f f i c i e n t c o n d i t i o n r e f e r r e d to i n the p r o o f above r e q u i r e s (RCC) and i t i s not known i f examples e x i s t of modules ( w i t h o u t (RCC)) over commutative r i n g s which have t e r t i a r y submodules which are not p r i m a r y . C. P r i m a l submodules. The i d e a of a p r i m a l submodule ( [ 1 ] , [5] and [9]) i s not a g e n e r a l i z a t i o n of the c l a s s i c a l p r i m a r y i d e a l s i n c e even i n commutative r i n g s w i t h the a s c e n d i n g c h a i n c o n d i t i o n thearc are p r i m a l i d e a l s which are not p r i m a r y ( P a r t IV, Example 3)» However, every p r i m a r y i d e a l , i s p r i m a l . In t h i s s e c t i o n the d e f i n i t i o n of p r i m a l submodules i s i n c l u d e d w i t h some ele m e n t a r y r e s u l t s because of t h e i r c l o s e c o n n e c t i o n w i t h o t h e r t o p i c s of t h i s t h e s i s . S e v e r a l r e s u l t s are quoted w i t h o u t p r o o f s . D e f i n i t i o n 34s A submodule L of a l e f t R-module M i s c a l l e d © p r i m a l i f L - s - z > L and L - J—A 2 D L imply t h a t (L-s-A^ n (L-s-A.) 3 L. C l e a r l y , f ) - i r r e d u c i b l e submodules are p r i m a l . 29 Theorem 35s In a module w i t h (RCC), a submodule L i s p r i m a l i f , and o n l y i f , i t has e x a c t l y one maximum p r o p e r l e f t r e s i d u a l P. P r o o f : I f P i s the unique maximum p r o p e r l e f t r e s i d u a l of L then L-s—A^ 3 L i m p l i e s A cr P by Lemma 9 of P a r t I , Hence, L - s — a L, L-s-A0 3 L im p l y A± cr JP, • A ' 2 cr P 2 and ^  + A 0 cr P. Thus, (L-s-i^) n (L - 8-A 2) = L*- ( A | * A g ) 3 L. C o n v e r s e l y , i f and P_2 are two d i s t i n c t maximal p r o p e r l e f t r e s i d u a l s of L, then L-s—Pj, 3 L and L-s-P_2 => L but (lU-P^ (L+r-Pg) = L - i - ^ + Pg) - L or + P 2 would be c o n t a i n e d ' i n a maximal p r o p e r l e f t r e s i d u a l of L. J | •• • / • In a module w i t h (RCC), a p r i m a l submodule L has a unique maximum p r o p e r l e f t r e s i d u a l J? which i s pr i m e . We c a l l L P - p r i m a l . P r i m a l submodules allow; d e c o m p o s i t i o n and uniqueness theorems analogous t o thos e f o r t e r t i a r y , s u b m o d u l e s . D e f i n i t i o n 36: I f a submodule L can be e x p r e s s e d as a f i n i t e i n t e r s e c t i o n of submodules, L = L^ L 2 n • • • n 1^, such t h a t the i n t e r s e c t i o n i s i r r e d u n d a n t and no submodule L^ can be r e p l a c e d by a r i g h t r e s i d u a l of L^ which s t r i c t l y c o n t a i n s L^ then the i n t e r s e c t i o n i s c a l l e d r educed. The next lemma i s the one r e f e r r e d t o f o l l o w i n g Theorem 28. Lemma 37: In a submodule w i t h (RCC), e v e r y n o n - p r i m a l submodule L has a r e p r e s e n t a t i o n as a reduced i n t e r s e c t i o n of two r i g h t r e s i d u a l s of L one of which i s p r i m a l . 30 P r o o f : I f L i s not p r i m a l t h e r e e x i s t s A j , A^ such t h a t L*-K1 D L, L+-A 2 z> L and L = (L-s-A.^ n (L-s-Ag). S i n c e the (RCC) g i v e s the a s c e n d i n g c h a i n c o n d i t i o n on l e f t r e s i d u a l s of L we have the d e s c e n d i n g c h a i n c o n d i t i o n on r i g h t r e s i d u a l s of L. To show t h i s l e t a be a d e s c e n d i n g c h a i n of r i g h t r e s i -d u a l s of L, = L-s—B^. C o n s i d e r the a s c e n d i n g c h a i n of l e f t r e s i d u a l s L—»L, cr L—sL_ cr • • • • For some n, L—»L = L—s-L A, ± — <. — n n • J. . = ••• . W e have B X , L A , c L so B x , c L—sL A. = L—sL , but —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 . Thus, t h e r e e x i s t s n n + 1 —n + 1 n — ' a r i g h t r e s i d u a l of L which i s m i n i m a l 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 L and L cr c L - t-Ag. The set of r i g h t r e s i d u a l s of L of the form X when X p L-s—A^ and L = X has a maximal element L^. I f L^ =» (L^-t—J3^) 0 ( L 1 " S — J 2 ) w i t h L^-s—_B^ r> L 1 and L^-s-J 2 Z) then (^-s - j p fl N 1 = H 1 and (L 1-*-B 2) n ^ = 1^ so t h a t 1^ cr L1 and L = which i s a c o n t r a d i c t i o n . Thus L^ must be p r i m a l . The set of r i g h t r e s i d u a l s Y of L s a t i s f y i n g Y rs H^ and Y n L^ = L has a maximal element L 2 so t h a t L = L^ n L 2 s a t i s f y i n g the lemma. B T h i s lemma may be used t o prove the f o l l o w i n g e x i s t e n c e theorem which i s g i v e n here w i t h o u t p r o o f . o Theorem 38: I f M has (RCC) then e v e r y submodule L f M. has a reduced i n t e r s e c t i o n of a f i n i t e number of p r i m a l submodules, L = L1 n ••• D L k# and i f L i i s j ^ - p r i m a l , ?± £ I! j # -Ej £ f o r a l l i ^ j . 31 Uniqueness f o l l o w s q u i t e r e a d i l y . Theorem 39: I f M has (RCC) and L f M has two reduced r e p r e s e n t a t i o n s as i n Theorem J>&, then the number of components i s the same f o r both and the s e t s of primes c o i n c i d e i n some o r d e r . The u n i q u e l y d e f i n e d set of primes i s the set of maximal p r o p e r l e f t r e s i d u a l s of L. D. The 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-By Theorems 16, 25 and 35 we can r e l a t e the d e f i n i t i o n s of p r i m a r y , t e r t i a r y and p r i m a l submodules t o p r o p e r l e f t r e s i d u a l s . I f M has (RCC) then any submodule L has f i n i t e l y many (1) p r o p e r prime l e f t r e s i d u a l s which are m i n i m a l i n the s e t of i d e a l s c o n t a i n i n g L—tM, (2) e s s e n t i a l r e s i d u a l s and (3) maximal p r o p e r l e f t r e s i d u a l s . So i f M has (RCC) we have the r e l a t i o n s h i p s i l l u s t r a t e d below: L p r i m a r y <£r^. L has e x a c t l y one p r o p e r prime l e f t modules. r e s i d u a l and i t i s minimum i n the s e t of i d e a l s c o n t a i n i n g L—tM. L t e r t i a r y <^ z> L has e x a c t l y one e s s e n t i a l r e s i d u a l . L p r i m a l L has e x a c t l y one maximal p r o p e r l e f t r e s i d u a l . 32 PART I I I G e n e r a l i z a t i o n s of the Second Uniqueness Theorem A» I s o l a t e d components. D e f i n i t i o n 1: Let A be an i d e a l of a ring; R and M a subset of K. The r i g h t upper M component of A i s the i n t e r s e c t i o n of a l l i d e a l s B c o n t a i n i n g A such that every element of M i s r i g h t prime to B. I f M i s the n u l l s e t , the component i s d e f i n e d to be A i t s e l f . The n o t a t i o n i s u(A, M). By d e f i n i t i o n , u(A, M) i s an i d e a l c o n t a i n i n g £ • Lemma 2; I f A c B a n d M n B » 0 , then u(A, M) c u(B, M) . P r o o f : I f X i s an i d e a l c o n t a i n i n g B f o r which every m £ M i s rp to X, then X i s such an i d e a l f o r A. 3 Lemma 3: I f and are sets such that 0 ^  1 H2 0 A • t and M1 c M 2 then u C A ^ ) C u(A, M 2). Pro o f : I f X i s an i d e a l c o n t a i n i n g A such that every m e i s rp to X then every m e i s rp to X, J| D e f i n i t i o n L: Let A be an i d e a l and M an m-system. I f H M we d e f i n e the r i g h t lower M component of A as (x e R: mfix £ A f o r some m e M). I f M » 0 we define the r i g h t lower M component as A i t s e l f . We use the n o t a t i o n t(h, M), In what f o l l o w s the case where M = 0 i s always o b v i o u s . Lemma 5 - For any i d e a l A and m-system M, i(A, M) i s an i d e a l * 33 P r o o f : I f x, y £ M) then f o r some m 0 £ M, m^Rx cr A, m 2 R y — —• F o r s o m e z £ R, m^zn^ £ M and m 1zm 2R(x - y) cr A. The remainder i s o b v i o u s . J| Lemma 6: For any i d e a l A and ra-system M, A c t ( A , M) cr u(A, M). P r o o f : A cr -t(A, M) i s o b v i o u s . Let X be an i d e a l such t h a t A cr X and e v e r y m £ M i s rp to X. Then, 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 E X. ] 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). P r o o f : (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). Let X be such t h a t A c X and m £ M i m p l i e s m i s rp to X, then X ro u(A, M) by d e f i n i t i o n and u(A, M) i s such an i d e a l . N o t i c e t h a t Lemma 2, Lemma 6 and (a) imply t h a t the o p e r a t i o n which c o r r e s p o n d s t o A the i d e a l u(A, M) i s a c l o s u r e o p e r a t i o n . S i n c e t h i s i s so, (b) f o l l o w s from Lemma 6. We have ,t[u(A, M), M] rp. u(A, M) but by Lemma 6, t [ u ( A , M), M] c u[u(A, M), MJ = u(A, M). Jj Theorem 8: I f R has the a s c e n d i n g c h a i n c o n d i t i o n f o r i d e a l s then f o r any i d e a l A and m-s ystem M, u(A, M) = *(A, M). T h i s i s a r e s u l t of Barnes* t o be found i n [12] (Theorem 1 2 ) . 34 The remainder of t h i s s e c t i o n w i l l be to give the a c t u a l g e n e r a l i z a t i o n of the second uniqueness theorem u s i n g the com-ponents d e f i n e d above. Lemma 9: If Q i s r i g h t primary, M an m-system contained i n the complement of r ( ^ ) , then Q = u( ^ , M). Proof : Q cr .£ and every m e M i s rp to Q. Theorem 10: I f A = f) 0 0 -2k i s a n irredundant primary r e p r e s e n t a t i o n of A with P^ = r ( ( ^ ) , then x i s rp to A i f x t P^ f o r i = 1, r . If the r i n g has the ascending chain c o n d i t i o n f o r i d e a l s the converse a l s o holds ( [ l l ] # Theorem 12). Theorem 11: (The Second Uniqueness Theorem) Let A = f l ^ g A A -^ k b e a n i r r e d u n d a n t primary r e p r e s e n t a -t i o n of A with r ( ^ i ) = I \ . If P i s a proper prime i d e a l which contains P,. P but does not con t a i n P ..., P,, then — 1 — r r+l —K U ( A , ^P) = £x n • • • n Q r-Proof: I f P D P, then u ( A , _P) cr u ( A , ^ . ) by Lemma 3. But, ^ i 2 A and every element of i s rp to j ^ . Thus, u ( A , ^P.) cr Hence, u ( A , ^ ) cr £± 0 ••• .fl 5 r - If r = k, U ( A , ^P) = A . If r < k and P does not con t a i n P j , j = r+1, k, then P does not c o n t a i n Q^t j = r + 1, k. This f o l l o w s s i n c e P. i s the unique minimum prime c o n t a i n i n g Q.. J J Thus, there are elements m^ , m^, ••«, ™ k _ r with £ Q r + i * m. t P. The elements ra, are i n the m-system ^JP so there e x i s t l — i ~" x 1 # «.., x k r _ 1 £ R such that m = n^x^n^x.^ x k _ r _ i m k _ r i s a n 35 element of ,^P. But, m £ £T + i 0 ••• D .Qk» So, i f q £ ^ fl ••• fl jJ # m R < J £ A. I f X i s an i d e a l , A r X, and n t P i m p l i e s n i s rp to X, then q £ fl ••• (1 J2R, mRq c A c X and t h i s i m p l i e s q £ X. Thus p ••• fl ^ f c u(A, ^ P) and the theorem f o l l o w s . J | Th i s i s not the form of the second uniqueness theorem g i v e n i n P a r t I but i f { P 1 # • • P , } i s an i s o l a t e d set of primes l e t Zl> •••» 2 h be the maximal primes i n the s e t . Then u(A, ^ £ 1 ) fl ••• nu(A, ^ £ h ) = .P^  D ••• fl Q. i s u n i q u e l y d e t e r m i n e d . C o r o l l a r y 12: I f R has the a s c e n d i n g c h a i n c o n d i t i o n f o r i d e a l s and i f A = ^  fl ••• D .Qk 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 i f P M i s a m i n i m a l prime c o n t a i n i n g A, then u(A, ^ P) i s r i g h t p r i m a r y and i s e q u a l to one of the i d e a l s jQ ,^ B. A g e n e r a l component f o r modules w i t h (RCC). D e f i n i t i o n 13: Let R be a r i n g and M a l e f t R-module w i t h (RCC). Then, f o r any submodule L and any prime i d e a l P H we d e f i n e the i s o l a t e d .P component of L, Lp, as the union of a l l sub-modules of the form L-t—B when _B gz P. D e f i n e L^ = L. The s e t of submodules L-s—B where _B c£ P has a maximal element by the (RCC). I f L-s—JB^  and L-s—B^ are two maximal elements (L-s—JB^) + (L-s—B^) CZ L - s — j ^ j ^ which i s a member of the set s i n c e P i s p r i m e . Thus Lp i s w e l l d e f i n e d . The (RCC) c o n d i t i o n i s assumed throughout the whole s e c t i o n and i s f r e q u e n t l y mentioned by i n s e r t i n g the symbol (RCC) at the s t a r t of the statement of a theorem. 36 Lemma 14: (RCC). I f L ez N and P i s prime then Lp c Np» P r o o f : I f L« = L t - B , B £ P, then BL» c L cr N so L» Q Nt-B c: N p . 3 Lemma 15 : (RCC). I f P^ e~ Pg are p r i m e s , then f o r any L, Lp 2 Lp • - 1 - 2 P r o o f : I f L» = L t - B , B £ ? 2 t h e n J £ -F-i a n d L* S L p • B Lemma 16: (RCC). F o r any L and prime P, [ L p ] p = Lp. P r o o f : C l e a r l y Lp 3 L so t h a t [ L p ] p 3 L p by Lemma 14. I f N = Lp-t-C, C £ P we wish t o show t h a t N cr L-t-C* f o r some C' £ P. Now, CN cr L p so f o r some C* £ P, CN ez L t - C * and N cr L-t—C*C ez L p where C*C £ P s i n c e P i s prime. 3 We may combine the above lemmas t o s t a t e Theorem 17: (RCC). The f u n c t i o n which maps L onto Lp f o r any f i x e d prime P i s a c l o s u r e o p e r a t i o n on the l a t t i c e o f sub-modules of M. A f u r t h e r p r o p e r t y of t h i s f u n c t i o n i s Theorem 18: (RCC). The f u n c t i o n which maps L onto L p f o r any f i x e d prime P i s an fl-endomorphism of the l a t t i c e of sub-modules of M. P r o o f : We must show t h a t L p f) N p - (L fl N ) p . We have L f l N c r L , L n N g N so by Lemma 14, (L n N ) p ez L p n N p . I f L t £ L p , N« ez N p then f o r some B £ P, C £ P, BL* ez L and 37 CN» cr N. Then BC cr (L* n N * ) Q L fl N. Thus L» n H » cr ( L n M ) P . J | We are now abl e t o c o n s i d e r the second uniqueness theorem i n the more g e n e r a l s e t t i n g of modules. Lemma 19: I f M has (RCC) and Q i s p r i m a r y and i f P i s a prime such t h a t r(Q) cr P then Q p = Q. P r o o f : I f S £ P then S £ r(Q) and i f L = Q-j-S then SL cr Q, S £ r(Q) so L c Q. So Q p c Q. ] . Theorem 2 0 : I f M has (RCC) and L = Q 1 n •«• 0 Qj, i s an i r r e d u n -dant p r i m a r y r e p r e s e n t a t i o n of L w i t h r(Q^) = P^ and i f ? i s a p r o p e r prime i d e a l which c o n t a i n s P^, P f but does not c o n t a i n P P, , then L n = Q, p • • • f> Q • —r+1*- —k P 1 r P r o o f : By Theorem 18, L p = Q^p 0 •••0 Qj, p so t h a t L p cz Q i p n 0 Q r P and by Lemma 1 9 , Q i p 0 • • • 0 Q f P = Q. n • " • fl Q • We need to show t h a t ("I ••• P Q r £ L p . S i n c e r n. " . — — i = r ^ i ^ » ^ o r s o m e ni» £ Q^ * C o n s i d e r - --n - n _ n, P r-> P in V ( « i 0 ••• 0 Q )• T h i s i s c o n t a i n e d i n — r + 1 — r + 2 —k 1 .^r Q 1 n • • • 0 Q p and i s c o n t a i n e d i n Q r +^ fi ••• 0. Q k s i n c e P / M C Q.. Thus P ^ * 1 P k k ( Q i n • • • n Q r) £ , L and N +1 n k P r ^ 1 Pfe £ P s i n c e P i s p r i m e . Hence Q 1 fl ••• D Q r £ L p . J I f [P^, P j } i s an i s o l a t e d s e t of p r i m e s , l e t P^, P^ be the maximal primes of t h i s s e t . Thus L p fl ••• 0 L p • -1 -h Q 1 fl ••• fl Qj which i s the second uniqueness theorem i n the form of P a r t I . 38 Theorem 21: I f M has (RCC) and L • (1 ••• (1 QFC 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 of L, then a prime P c o n t a i n i n g L—tM i s one of the primes P^ = r ( Q ^ ) , i = 1, • k, i f , and o n l y i f , Lp*—P 3 L p . P r o o f : S i n c e r ( L ) = r ( Q 1 ) n • • • Tl r (Q f c ) = ? 1 fl ••• f! P f c i t i s c l e a r t h a t the primes P^, •'"*/ Pj. i n c l u d e the m i n i m a l ; primes c o n t a i n i n g L—tM. ( i ) Let P = P . Then P 3 P^, • P r w i t h P ± p r + 1 * Then L p = 0^ n . . .- n Q and t h i s 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 of Lp, Lp*—A 3 Lp i f , and o n l y i f , A c P j ez P f o r some j , 1 < j < r (Lemma 18, P a r t I I ) and, t h e r e -f o r e , Lp+-P 3 Lp. ( i i ) Assume P 3 L—sM i s a prime and P i s such t h a t Lp-s—JP 3 Lp. S i n c e the m i n i m a l primes c o n t a i n i n g L—tM are among the P^, ••», P^, P must c o n t a i n one of them so assume 1 2. P X f "••# lz a n d P ± lT+1> •"•# P k- T h e n L p = Q x n • • • 0 Q r 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 of Lp. S i n c e Lp-t—P 3 Lp, Lemma 18 of P a r t I I g i v e s t h a t f o r some i , 1 < i < r , P ez P^. ] As an immediate c o r o l l a r y we have Theorem 20 of P a r t I I or the f i r s t u niqueness theorem f o r p r i m a r y r e p r e s e n t a t i o n s s i n c e the prime r a d i c a l s i n a s h o r t r e p r e s e n t a t i o n are not dependent on the p a r t i c u l a r p r i m a r y modules of the r e p r e s e n t a t i o n . The p r o o f s above can be g e n e r a l i z e d i m m e d i a t e l y t o the m u l t i p l i c a t i v e l a t t i c e s of [9] and i n p a r t i c u l a r t o the case where L, N, ••• and A, JB, ••• are a l l t w o - s i d e d i d e a l s of the same r i n g . With the ob v i o u s n o t a t i o n we have 39 Theorem 22: For any r i n g R w i t h (RCC), and any i d e a l A and prime i d e a l P, A p = ^(A, ,^P). P r o o f : I f x e -l(A, ^ ) then f o r some m t P, aRx c A. Hence, raR(x) gz A and (m) (x) cr A, (m) £ P so (x) cr A p and x e A p . I f x e A p then f o r some C £ P, x e At—C. But, A-J—C = [x: Cx <- AJ cr (x: .CRx cr A} SO f o r some m e C fl ~£_t niRx cr A and x e -t(A, ^ P ) . 2 The next s t e p would be t o f i n d analogous r e s u l t s f o r t e r t i a r y r e p r e s e n t a t i o n s , however i n t h i s case much l e s s may be s t a t e d . There i s a uniqueness theorem f o r p r i m a l r e p r e s e n t a t i o n s which i s analogous t o t h e second uniqueness theorem. Lemma 23: In a module w i t h (RCC), i f L = ^ fl ••• p. L^ (k > 2) i s a reduced p r i m a l r e p r e s e n t a t i o n of L and i f the unique maximum p r o p e r l e f t r e s i d u a l of L, i s P. then L_ i s P.— l — i P^ — l p r i m a l and L p c L,, L = ^ fl ••• fl fl L p n L i + 1 — i — i fl ••• fl L^ i s a reduced p r i m a l r e p r e s e n t a t i o n . P r o o f : We w i l l prove the r e s u l t f o r i = 1. By Theorem 18, L p = L^p n ••• fl L^p . S i n c e P^ i s the maximum p r o p e r l e f t r e s i d u a l of L^, we have L^p = L^. Thus L cr L p c L^t and L = L p fl L 2 fl ••• fl L^. Next, every p r o p e r l e f t r e s i d u a l of Lp i s c o n t a i n e d i n P^. I f n o t , we have A £ P^ and L p +-A rr> L p . Then f o r some B £ P, , B A ( L p +-A) cr L, BA £ P - 1 - 1 -1 1 s i n c e P, i s prime and so L p -s—A cr L p which i s a c o n t r a d i c t i o n . 1 - 1 - 1 40 Thus A £ V i m p l i e s L p -s-A = L p . We next show t h a t P, i s a -1 -1 1 maximal p r o p e r l e f t r e s i d u a l of L p . By the above, i t i s enough -1 . to show t h a t P. i s a p r o p e r l e f t r e s i d u a l of L p . To show t h i s , -1 note t h a t L p p L 2 n • • • 0 L f c = L cr L+-P-J = ( L p i-R^ 0 f| ••• fl Lj, so Lp * — P j Lp and P^ i s a p r o p e r l e f t r e s i d u a l of L p . Hence, L p i s P , - p r i m a l . 1 1 We must now show t h a t L = Lp (1 ^ D *•• H ^ i s reduced. Replace one component, say L^, by a r i g h t r e s i d u a l which i s s t r i c t l y l a r g e r than L^. So assume L = Lp 0 (Lg-s—A) f| ••• fl L^ where L^*—A 3 L^. S i n c e i s P g - p r i r a a l , A cr .£ 2 a n d L = Lp n (Lg-s—P 2) fi ••• fl Lj, s L-s—P 2 which i s a c o n t r a d i c t i o n s i n c e Pg i s a p r o p e r l e f t r e s i d u a l of L. S i m i l a r l y f o r any component. J| C o r o l l a r y 24: In a module w i t h (RCC), i f L = L^ f) fl L^ i s a reduced p r i m a l r e p r e s e n t a t i o n of L where L^ i s P ^ - p r i m a l , Lp i s the s m a l l e s t j ^ . - p r i m a l submodule which may r e p l a c e L. i n the r e p r e s e n t a t i o n . By a p p l y i n g the method of Lemma 23 k t i m e s we have Theorem 25: In a module w i t h (RCC) i f L i s a submodule w i t h maximal p r o p e r l e f t r e s i d u a l s P_^ , P^, k > 1, then L = L p 0 L p Q ••• n L p i s a reduced p r i m a l r e p r e s e n t a t i o n -1 -2 -k I f L = n n i s any reduced p r i m a l r e p r e s e n t a t i o n w i t h the components l a b e l l e d so t h a t 1^ i s P ^ - p r i r a a l then - i 1 T h i s 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 d e c o m p o s i t i o n . Lemma 23 may be weakened somewhat as f o l l o w s . Theorem 26; I f M has (RCC) and L i s a submodule o f M then i f 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 . P r o o f : Let L - fl • • • 0. T f c be a reduced t e r t i a r y r e p r e s e n t a t i o n of L and assume t h a t i s P ^ - t e r t i a r y and t h a t P x - P. Now, L p ^ = T i p ^ n T 2 p ^ n ••• fl T k p ^ . But, P_x i s the maximum p r o p e r l e f t r e s i d u a l of T^ so T^p «= T^. Hence, Lp - T1 n T 2 p f) ••• fl T k p and i s not redundant i n t h i s ~ 1 ~ 1 ~ 1 k k i n t e r s e c t i o n s i n c e i f i t were, 2 (~\ T. p 2 f^] T^ i=2 —1 i a 2 which c o n t r a d i c t s the i r r e d u n d a n c y of the r e p r e s e n t a t i o n o f L. Hence, t h e r e i s a reduced t e r t i a r y r e p r e s e n t a t i o n of L p w i t h - 1 a P ^ - t e r t i a r y component. That i s , P^ i s an e s s e n t i a l r e s i d u a l of L p . But, A £ P., i m p l i e s L p -j—A = L p so P, i s the maximum -1 1 -1 -1 1 p r o p e r l e f t r e s i d u a l of L p . T h e r e f o r e , L p i s P . - p r i m a l . ] | -1 -1 1 U2 PART IV Examples and Remarks Example 1; The (RCC) c o n d i t i o n i s not s u f f i c i e n t to guarantee the e x i s t e n c e of primary r e p r e s e n t a t i o n s . The f o l l o w i n g example ([2]) i s a noncommutative r i n g with the ascending chain c o n d i t i o n . Let K be a f i e l d and § the algebra over K generated by e l ' e 2 ' n w ^ e r e e i = e i * e2 = e2* n ^ = ®' e l e 2 = e 2 e l = ^' e^n ne n = n, ,n = ne, = 0. The i d e a l s of $ are c l e a r l y (0), ( e 1 ) , ( e 2 ) ' ^ n ^ ' ^ e i ' e 2 ^ a n d T h e P r i m e s a r e ( ei» e 2 ) » ^ e i ) and ( e 2)» The minimal primes are (e^) and ( e 2 ) so that r(0) = ( e ; L) n ( e 2 ) = (n). But, (e^ie^ c (n) so (n) i s not prime. Hence (0) i s not primary. But, (0) i s fl-irreducible'and so does not have a primary r e p r e s e n t a t i o n . The i d e a l (0) ( , e l ' e 2 } ( e x ) ( e ? ) (n) (0) i s t e r t i a r y and ( e 2 ) * s e s s e n t i a l r e s i d u a l of (0) s i n c e ( O ) - * ^ ) = {x: x ( e i ) = 0} = ( e 2 ) and (O)-e-(n) = [x: x(n) = 0} = ( e 2 ) . To emphasize the d i f f e r e n c e between primary and t e r t i a r y i d e a l s the f o l l o w i n g i s a f i n i t e example with a t e r t i a r y i d e a l which i s not primary. Example 2; Consider the r i n g given by the f o l l o w i n g t a b l e s : 1 a b c d e f 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 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 a d c b e 1 43 That t h i s i s a r i n g i s shown by the f a c t t h a t the t a b l e s are s a t i s f i e d by the f o l l o w i n g m a t r i c e s over a f i e l d of c h a r a c t e r i s t i c two. The zer o and 1 elements are the zero and i d e n t i t y m a t r i c e s . /0 0 1 \ / 1 0 0\ a - I 0 0 0 ) , b = I O 0 0 J , c = \0 0 0/ \0 0 0/ / 1 0 1 \ / 0 0 0 \ / 1 0 1 \ d = 0 0 0 , e = 0 1 0 ) , f = ( 0 1 0 ) . V o o o / \ 0 0 1 / \0 0 1./ The t w o - s i d e d i d e a l s a r e : (0), (a) = [0, a}, (b) = (d) - ( a , b) = {0, a, b, d},' (c) = (e) - ( a , c) = fO, a, c, e } f R. 2 S i n c e R has a u n i t , R • R, R(a, b) = (a , b)R = ( a , b) so ( a , c) i s pr i m e . ( a , b ) ^ ^)(a,c) S i m i l a r l y , ( a , b) i s pr i m e . To compute \(0) we need the maximum i d e a l A such t h a t ((O)-t-A) n X = (0) i m p l i e s X = (0). But, (O)-t-(a) = {x: ( a ) x = 0} = ( a , b) which s a t i s f i e s the requirement and (0)-s—(a, c) = (x: ax = cx = ex = o} = ( a , b ) . S i n c e (O)-j-R = (0) and (O)-j-(a, b) = {x: ax = bx = dx = 0} » (0), X(0) = ( a , c ) . To f i n d \ ( ( a ) ) we note (a)-*-(a, b) = ( a , c) but ( a , c) n ( a , b) = (a) and (a, 1 b) £ ( a ) . A l s o , ( a ) t - ( a , c) = (a,b) and ( a , b) fl ( a , c) = (a) as b e f o r e and ( a , c) £ ( a ) . F i n a l l y , then A.((a)) = ( a ) . The p r i m a r y r a d i c a l s can be seen t o be r(0) = (a) and r ( ( a ) ) = ( a ) . S i n c e (a) i s not p r i m e , (0) i s not p r i m a r y . T h i s example may a l s o be c o n s i d e r e d as a l e f t R-module. We have ( a | = [0, a}, (b| = ( a , b| - (d| = {0,a,b,d}, ( c j = [0, c } , ( a , c| = {0, a, e, e}, (ej = [Q, e } , R. I t can 44 be shown t h a t \(0) = ( a , c ) , ?vj((a|) • ( a ) , r(0) = (a) and r ( ( a | ) - ( a ) . Example 3: The f o l l o w i n g example i s a commutative r i n g w i t h a s c e n d i n g c h a i n c o n d i t i o n i n which t h e r e are p r i m a l i d e a l s which are not p r i m a r y . Let K [ x , y, z] be the p o l y n o m i a l r i n g i n t h r e e commutative v a r i a b l e s o ver a f i e l d K. C o n s i d e r the i d e a l 2 2 I = (x y, xy ). T h i s i d e a l i s not p r i m a r y s i n c e 2 2 ( x , y ) ( x y ) c (x y, xy ) and (xy) I and no power of (x, y) i s c o n t a i n e d i n I . .However, i t i s easy t o see t h a t (x, y) i s the maximum p r o p e r l e f t r e s i d u a l of I . Remark 4: 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 i n terras of elements of a r i n g o r module. The d e f i n i t i o n s g i v e n below are e a s i l y seen t o be e q u i v a l e n t t o the one g i v e n i n P a r t I I but o b v i o u s l y are w e l l d e f i n e d even i f t h e r e i s no (RCC) c o n d i t i o n . D e f i n i t i o n i In a r i n g R, the t e r t i a r y r a d i c a l of a l e f t i d e a l I i s the s e t of elements a £ R such t h a t b % I i m p l i e s t h e r e e x i s t s x £ (b| such t h a t x t I and aR*x cz I (aR*x = aRx (j [ a x } ) . In a r i n g R, the t e r t i a r y r a d i c a l of a ( t w o - s i d e d ) i d e a l _I i s the s e t of elements a £ R such t h a t h t I i m p l i e s t h e r e e x i s t s x £ (b) such t h a t x $ I and aR*x cz I» Any t w o - s i d e d i d e a l has two t e r t i a r y r a d i c a l s — one when c o n s i d e r e d as a l e f t i d e a l and when c o n s i d e r e d as an i d e a l . S i n c e (b| cz (b) the r a d i c a l of the l e f t i d e a l i s c o n t a i n e d i n the r a d i c a l of the i d e a l . I t can be p r o v e d ([6], p. 469) t h a t the two r a d i c a l s c o i n c i d e i f the r i n g has the d e s c e n d i n g c h a i n c o n d i t i o n on l e f t i d e a l s . 45 Remark 5 i Other connections between r e l a t e d i d e a l and module s t r u c t u r e s have not been f u l l y i n v e s t i g a t e d . As an example, i f M i s a l e f t R-module with (RCC) and i f the l a t t i c e of two-sided i d e a l s of R also has (RCC) then both a submodule L and the two-s i d e d i d e a l L—tM have t e r t i a r y r e p r e s e n t a t i o n s . The f o l l o w i n g simple theorem gives some r e l a t i o n s h i p s between the two. Theorem: If M i s a l e f t R-module with (RCC) and the l a t t i c e of two-sided i d e a l s of R has (RCC), then any proper l e f t r e s i d u a l of the i d e a l L—*M i s contained i n a proper l e f t r e s i d u a l of the submodule L. In p a r t i c u l a r , X(L—*M) ez A.(L). Proof: Let L—tM = L and B = L—tA, A £ L, a proper l e f t r e s i d u a l of _L. The i d e a l _B i s the sum of a l l i d e a l s C such that C A c L = L—tM or CAM cz L. But, s i n c e A £ L = L—tM, AM £ L and L—tAM i s a proper l e f t r e s i d u a l of L. Hence, _B = _L—tA c L—tAM. 3 C o r o l l a r y : Under the assumptions of the theorem, i f L = T^ 0 ••• fl Ij, i s a t e r t i a r y r e p r e s e n t a t i o n with A-(T^) = P^ and i f L—tM = T^ p • • • 0 T^ i s a t e r t i a r y r e p r e s e n t a t i o n with A.(T.) = P! then each P! cz P. f o r some j . - i - i - i j Remark 6: In t h i s remark some well-known p r o p e r t i e s of r i n g s are reworded to correspond to the language of t h i s t h e s i s . Every maximal l e f t i d e a l i s t e r t i a r y s i n c e maximal l e f t i d e a l s are c l e a r l y P - i r r e d u c i b l e . I f T i s a maximal l e f t i d e a l , c onsider X,(T) = {a: b & T i m p l i e s there e x i s t s x £ (b| such that x t T and aR*x c TJ and T-*R = f a : aR ez T J . If a e T—tR, c l e a r l y a £ A.(T). If a £ A.(T) f o r every b I T there i s an 46 x e (b| such t h a t x t T and aR*x c T so aR*(x| cz T, hence a(T + (x|) = aR c T and a e T—sR. Hence X,(T) = T—*R. Note t h a t T—s-R i s t r i v i a l l y an e s s e n t i a l r e s i d u a l of T s i n c e t h e r e are no i n t e r m e d i a t e l e f t i d e a l s between T and R. By Theorem 14, P a r t I , every e s s e n t i a l r e s i d u a l i s prime (no c h a i n c o n d i t i o n i s r e q u i r e d ) and so T—s-R i s p r i m e . I t i s w e l l known t h a t an i d e a l P i s p r i m i t i v e i n R ( i . e . , R/P i s a p r i m i t i v e r i n g ) i f P = L—*R where L i s a modular maximal l e f t i d e a l . Thus P i s p r i m i t i v e i f i t i s the e s s e n t i a l r e s i d u a l of a modular maximal l e f t i d e a l and so i s p r i m e . A r i n g R i s p r i m i t i v e i f i t has a maximal l e f t i d e a l I such t h a t I—s-R = 0. Thus a r i n g i s p r i m i t i v e i f the zero i d e a l i s the t e r t i a r y r a d i c a l of some maximal l e f t i d e a l . Remark 7: In Chapter V I I I of [9]» L e s i e u r and C r o i s o t have g i v e n r e s u l t s on t e r t i a r y d e c o m p o s i t i o n s i n r i n g s w i t h the d e s c e n d i n g c h a i n c o n d i t i o n on l e f t i d e a l s . These r e s u l t s are s t r o n g e r than those p o s s i b l e w i t h the (RCC) c o n d i t i o n . S i m i l a r r e s u l t s are not yet a v a i l a b l e f o r the case of r i n g s w i t h maximum c o n d i t i o n on l e f t i d e a l s o r i n the more g e n e r a l case of f i n i t e d i m e n s i o n a l modules. The f o l l o w i n g g e n e r a l theorem g i v e s r i s e t o a s i m p l e r e s u l t i n t h i s d i r e c t i o n . Theorem: Let R have (RCC) as a l e f t R-module and i f T i s P - t e r t i a r y then a £ R such t h a t Ra c£ T i m p l i e s T—sa i s P - t e r t i a r y . P r o o f : The r e s i d u a l T — t a i s c l e a r l y a l e f t i d e a l . We show 47 f i r s t t h a t A.(T—ta) = P. Let r £ P and b t T—*a, then ba t T. There e x i s t s x £ (ba| such t h a t x t T and r ( x ) ez T. S i n c e x £ ( b a | f x = ca f o r some c £ (b|. Thus c t T — t a and r ( c | ez T—»a. Hence r £ A.(T—ta). C o n v e r s e l y , i f r £ \(T—ta) then s i n c e Ra £ T, t h e r e i s some x t T — t a . Hence, t h e r e i s some x* £ (x| such t h a t r ( x f | cz T — t a . Then r ( x * a | cz T ( s i n c e r ( x * a | Q r ( x * | s ) and x»a t T. But, s i n c e T i s P - t e r t i a r y , P i s t h e maximum p r o p e r l e f t r e s i d u a l of T so r £ T — t ( x f a | ez P. We next show t h a t b ( c | c T — t a , c t T — t a i m p l i e s b £ P = 7v.(T-ta). I f b ( c | ez T — t a , c t T — t a then ca i T and b ( c a | cz T, but s i n c e T i s P - t e r t i a r y , b £ P. I t f o l l o w s t h a t i f A and B are l e f t i d e a l s , B ± T — t a , AB ez T — t a then A cz P and so T — t a i s P - t e r t i a r y . J The f o l l o w i n g c o r o l l a r y i s immediate ( i t i s t r u e a l s o i n the case of minimum c o n d i t i o n [9])« C o r o l l a r y : I f R i s c o n s i d e r e d as a l e f t R-module and R has the maximum c o n d i t i o n on l e f t i d e a l s then i f T i s P - t e r t i a r y and L i s a l e f t i d e a l such t h a t RL £ T then T — t L i s P - t e r t i a r y as a l e f t i d e a l . P r o o f : L i s f i n i t e l y g e n e r a t e d so L = ( a , , *'*, a | and the i n t e r s e c t i o n so we may assume t h a t Ra^ £ T and so by the theorem each T—ta^^^ i s P - t e r t i a r y . Thus Lemma 26 of P a r t I I g i v e s the r e s u l t . J | n i = l I f Ra, ez T then T — t a , = R i s s u p e r f l u o u s i n 48 References 1. W. E. Barnes. P r i m a l i d e a l s and i s o l a t e d components in non-commutative r i n g s . Trans. Amer. Math. S o c , 8J2 (1956), 1-16. 2. C. W. C u r t i s , .On a d d i t i v e i d e a l theory i n general r i n g s , Amer. Jour, of Math., 1 4 ( 1 9 5 2 ) , 687-700. 3. L. Fuchs, .On p r i m a l i d e a l s , Proc. Amer. Math. S o c , 1 (1950), 1-8. 4. A. W. G o l d i e , Rings with maximum c o n d i t i o n , (Yale U n i v e r s i t y , 1961, mimeographed n o t e s ) . 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 noetherienne des anneaux, des demi-groupes et des modules dans l e cas non commutatif I, C o l l . d'Alg. Sup., C.B.R.M. (1956"), 79 -121 . 6 . , I b i d . , I I , Math. Annalen, 134 (1958), 458-476. 7 . , I b i d . . I l l , Acad. Royale de Belgique, .44 (1958), 75-93-8. , La n o t i o n de r e s i d u a l e s s e n t i e l , Comptes Rendus Acad. Sc., .246 (1958), 357-360. 9 . Algebre noetherienne non commut at i ve ( P a r i s , 1963). 10. N, H, McCoy, Prime i de a l s i n general r i n g s , Amer. Jour, of Math., 21 (1949), 823 -833-11 . D. C. Murdoch, Cont f i but i on s to noncommut at i ve i d e a l theory, Can. Jour, of Math., .4 (1952), 43-57. 12. , Subrings of the maximal r i n g of q u o t i e n t s a s s o c i a t e d • with c l o s u r e o p e r a t i o n s . Can. Jour. of Math., .JJ5 (1963), 723-743. 

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