UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Topics on noncommutative localization and group rings Lee, Kit-sum 1978

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Notice for Google Chrome users:
If you are having trouble viewing or searching the PDF with Google Chrome, please download it here instead.

Item Metadata

Download

Media
831-UBC_1979_A1 L44.pdf [ 3.82MB ]
Metadata
JSON: 831-1.0080276.json
JSON-LD: 831-1.0080276-ld.json
RDF/XML (Pretty): 831-1.0080276-rdf.xml
RDF/JSON: 831-1.0080276-rdf.json
Turtle: 831-1.0080276-turtle.txt
N-Triples: 831-1.0080276-rdf-ntriples.txt
Original Record: 831-1.0080276-source.json
Full Text
831-1.0080276-fulltext.txt
Citation
831-1.0080276.ris

Full Text

TOPICS 08 NQNCOHMUTATIVB LOCALIZATION AND GROUP RINGS by KIT-SUM |LEE B . S c , The Chinese U n i v e r s i t y of Hong Kong, 1968 M.A., U n i v e r s i t y of New Brunswick, 1973 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENT FOR THE DEGREE OF D0CT08 Of PHILOSOPHY in T H E FACULTY OF GRADUATE STUDIES ( i n the Department of MATHEMATICS) BE &CCEPT THIS THESIS AS CONFORMING TO THE REQUIRED STANDARD THE UNIVERSITY OF BRITISH COLUMBIA NOVEMBER 1978 (^KIT-SUM L E E , 1978 ) 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 o f t h e r e q u i r e m e n t s " f o r a n a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l m a k e 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 a n d s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s m a y b e g r a n t e d b y t h e H e a d o f my D e p a r t m e n t o r b y h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s 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 n o t b e a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t o f M a t h e m a t i c s T h e U n i v e r s i t y o f B r i t i s h C o l u m b i a 2 0 7 5 W e s b r o o k P l a c e V a n c o u v e r , C a n a d a V 6 T 1W5 March 16 , , 1 9 7 9 ABSTRACT Supervisor: S. s . Page Three t o p i c s i n l o c a l i z a t i o n theory and group r i n g theory are i n v e s t i g a t e d . In Chapter I , i t i s proved t h a t every symmetric k e r n e l functor i n a l e f t Noetherian r i n g i s induced by a prime i d e a l . A s u f f i c i e n t c o n d i t i o n f o r the r i n g of q u o t i e n t s with respect t o a prime k e r n e l f u n c t o r to be semi-simple A r t i n i a n i s found. &n analogous r e s u l t f o r guasi-prime k e r n e l functors i s obtained i n Section 5. In Chapter I I , the idea of c o n t r o l l i n g subgroups i s a p p l i e d to group r i n g l o c a l i z a t i o n . Some s u f f i c i e n t c o n d i t i o n s f o r the descent of the maximal r i n g of q u o t i e n t s and of the c l a s s i c a l r i n g of quotients are obtained i n Section 8. In Chapter I I I , c h a r a c t e r i z a t i o n s of group r i n g s , over n i l p o t e n t groups of t r a n s f i n i t e l y bounded c a r d i n a l i t y , whose l e f t g l o b a l dimension i s f i n i t e are obtained. As an a p p l i c a t i o n , t h i s homological r e s u l t i s used to get i n f o r m a t i o n on the t o r s i o n elements of an M-group. i i i TABLE OF CONTENTS CHAPTER I. NONCOMMOTATI7E LOCALIZATION - . 1 Sect i o n 1. Torsion Theory .......................... 1. Sect i o n 2. Rings and Modules of Quotients .......... 7 Section 3. Examples ................................12 Sect i o n 4. Prime Kernel Functor 17 Section 5. Quasi-prime Kernel Functors ............. 25 CHAPTER I I . LOCALIZATION OF GROUP RINGS ..... 34 Sect i o n 6. The C o n t r o l l e r .......................... 34 Se c t i o n 7. The Rings Of Quotients Of A Group Bing ..40 Sec t i o n 8. The Maximal Sing Of Quotients And The C l a s s i c a l King Of Quotients 48 CHAPTER I I I . GLOBAL DIMENSION OF GfiOUP RINGS 53 Sect i o n 9. Background .............................. 53 Section 10. The Commutative Case ................... 59 Section 11. The F i n i t e Case And The Case Of M-gr oups ...... - - ...... ............... ... . -- .... . - »•, 61 Section 12. Osofsky*s Theorem And G e n e r a l i z a t i o n To L o c a l l y M-Groups ................................ 66 REFERENCES. . . . . , . . . . . . . . 7 2 ACKNOWLEDGEMENTS The author wishes to express h i s thanks to h i s su p e r v i s o r , Dr- S.S- Page, f o r encouragement and advice tendered during the execution of part of the research h e r e i n described. The author wishes t o thank Dr. D.C- Murdoch, h i s former s u p e r v i s o r , f o r a s s i s t a n c e during the execution of most of the research described i n Chapter I . The f i n a n c i a l support of the N a t i o n a l Research C o u n c i l of Canada i s g r a t e f u l l y acknowledged. 1 CHAPTEfi I NQNCOaaUTATIVE LOCALIZATION S e c t i o n J . Torsion Theory In t h i s s e c t i o n we s h a l l o u t l i n e the b a s i c f a c t s of t o r s i o n theory. & t o r s i o n theory i s c h a r a c t e r i z e d i n f o u r e q u i v a l e n t ways: (1). by a c l a s s of t o r s i o n modules; (2) by a c l a s s of t o r s i o n - f r e e modules; (.3) by an idempotent k e r n e l f u n c t o r ; (H) by a topology of l e f t i d e a l s . Every t o r s i o n theory i s generated by a module and cogeaerated by a module. The set of a l l t o r s i o n t h e o r i e s (with respect t o a f i x e d ring) i s equipped with a p a r t i a l order which t u r n s i t i n t o a complete l a t t i c e . Let B be an a s s o c i a t i v e r i n g with u n i t y . The category of l e f t u n i t a r y fi-modules i s denoted by ^ H , or B i f there i s no danger of confusion. For M, N i n M, the ab e l i a n group Hom/(?(M,N) i s denoted by £M,N1^, or £M,NJ. I(M) denotes the i n j e c t i v e h u l l of H i n H. To say f i n [M,NJ i s a monomorphism, we simply w r i t e f:M>—>N. S i m i l a r l y , f:H—>>N means f i n £M,Nj i s aa epimorphism..,. Let T and F be subclasses of H and H be i n M. By £a,F]=0, we mean £H,F.]=0 f o r every F i n F. By £M,I(F) j=0, we mean £M,I{F)]=0 f o r every F i n F. £T#H]=0 i s t r a n s l a t e d i n a s i m i l a r way. a 2 subclass C of M i s s a i d to be closed under extension i f f o r every short exact sequence M > — > N — » L i n M, M and L are i n C i m p l i e s N i s i n C. Throughout t h i s paper, the word " i d e a l " w i l l mean two sided i d e a l , unless s p e c i f i e d by " l e f t " or " r i g h t " ; the word "module" r e f e r s to l e f t module. 1.1. P r o p o s i t i o n . £20, P r o p o s i t i o n 0.3} Let T and F be subclasses of M, closed under isomorphic images. The f o l l o w i n g c o n d i t i o n s are equ i v a l e n t : (1) T i s closed under q u o t i e n t s , extensions, d i r e c t sums and submodules; and .£= {M |[T, M j=0] . (2) F i s closed under submodules, extensions, d i r e c t products and i n f e c t i v e h u l l s ; and T= {M | [ M ,1 (F) j=0j . (3) T={M1£M,I(F) >0} ; F= £ H | [ T > I ( H ) ]=0j. 1. 2. _Def i n i t i o n . A torsion_theory, i s a p a i r of subclasses (T,F) s a t i s f y i n g the c o n d i t i o n s i n P r o p o s i t i o n 1.1. Let (T,F) be a t o r s i o n theory, and M be i n M. Define K as the sum of a l l submodules of M which belong to T, and L as the i n t e r s e c t i o n of a l l submodules N of M such th a t M/N belongs t o F. Then K=L £20, P r o p o s i t i o n 0.3].., 1.3 D e f i n i t i o n . An idempptent k e r n e l f u n c t o r (abbreviated as IKF) i s a subfunctor t of the i d e n t i t y f u n c t o r of M such t h a t : (1) t(M/t(M))=0 f o r every H i n M, and (2) i f M<N i n H, 3 then t (M) =M f) t (N). I n other words, t assigns t o every M i n M a submodule t(M) and to every M, N i n M a map £ ia, N J —>[ t <M) , t (H) J by r e s t r i c t i o n , such that (1) and (2) are s a t i s f i e d . 1.4. P r o p o s i t i o n . {34, C o r o l l a r y 2.7] There i s an one-one correspondence between t o r s i o n t h e o r i e s and IKF*s. Let (T,F) be a t o r s i o n theory, the corresponding IKF t i s defined by : t(M) = the i n t e r s e c t i o n of a l l submodules N of fl such t h a t M/N i s i n F = the sum of a l l the submodules which belong to T. Conversely, to an IKF t there corresponds a t o r s i o n theory (T,F) such t h a t M i s i n T i f and only i f t(M)=H, and M i s i n F i f and only i f t(fl)=0. 1.5. D e f i n i t i o n . A topology i s a non-empty f a m i l y D of l e f t i d e a l s of fi s a t i s f y i n g : (1) I f A i s i n D, x i s i n R, then (A:x j i s i n D, where {.A:x]={r|r i n R, rx i n A j . (2) I f A i s i n D, B i s a l e f t i d e a l such t h a t £B:aJ i s i n D f o r every a i n A, then B i s i n D. I t can r e a d i l y be proved that a topology D a l s o s a t i s f i e s the f o l l o w i n g p r o p e r t i e s : (3) I f A i s i n D and B i s a l e f t i d e a l c ontaining A, then B i s i n D. (4) I f A, B are i n D, then the i n t e r s e c t i o n and the 4 product of A and B are i n D. When (1) , (3) and (4) hold, B i s a t o p o l o g i c a l r i n g with D as a fundamental system of neighborhoods of the zero element. 1.6..^proposition. £34, Theorem 3.4] There i s an one-one correspondence between IKF's and t o p o l o g i e s . The correspondence assigns t o an IKF t the topology D = {D J t(B/D)= B/Dj, and to a topology D the IKF t such t h a t t(M) equals the s e t of elements i n H a n n i h i l a t e d by some l e f t i d e a l i n D. Thus there i s a one-one correspondence among t o r s i o n t h e o r i e s , IKF*s and t o p o l o g i e s . Let t be an IKF, (T,F) and D be the corresponding t o r s i o n theory and the corresponding topology, r e s p e c t i v e l y . A module M i n M i s t - t o r s i o n i f i t i s i n T, t - t o r s i o n - f r e e i f i t i s i n F. A submodule N of M i s t-dense i f M/N i s t - t o r s i o n , and i s t - c l o s e d i f H/S i s t -t o r s i o n - f r e e . Thus D i s e x a c t l y the f a m i l y of a l l t-dense l e f t i d e a l s . 1.7. P r o p o s i t i o n . £10, P r o p o s i t i o n 7.1 J For 1=1,2, l e t the IKF^s t ( i ) correspond t o the t o r s i o n t h e o r i e s (T(i) ,F ( i | > and the t o p o l o g i e s D ( i ) . The f o l l o w i n g c o n d i t i o n s are equ i v a l e n t ; (1) - -HIL i s contained i n Tj[2}_. (2) l l l l . c o n tains F 12) . 5 (3) For each M i n M, t(1) <M) i s contained i n t{2) (13). M i l i s contained i n U.12.L- \ 1.8. . D e f i n i t i o n . The IKF t(1) i s smaller than t(2) , or t(2) i s greater than t ( 1 ) , or tlll.£%12L * i f a n d only i f the c o n d i t i o n s i n P r o p o s i t i o n 1.8 are s a t i s f i e d . 1.9. D e f i n i t i o n . Let C be a subclass of M. The s m a l l e s t t o r s i o n theory f o r which C i s contained i n T i s c a l l e d the t o r s i o n theory generated by C. The gr e a t e s t t o r s i o n theory f o r which C i s contained i n F i s c a l l e d the t o r s i o n theory co generated by C. Do such t o r s i o n t h e o r i e s e x i s t ? A l s o , i f they e x i s t , are they unique? Both questions are answered a f f i r m a t i v e l y i n the f o l l o w i n g way £20, Section Oj; For a subclass X of M, l e t 1 (X) = [M|£ M,I (X) ]=0j and r(X) = {H|1X,I(H) 3=0j. Then obviously lr{X) con t a i n s X, r l (X) c o n t a i n s X, and X i s contained i n Y i m p l i e s r (X) contains r (1) and 1{X) con t a i n s 1 (Y) . Let C be a subclass of H. Then r l r (C) =r ( l r (C)) i s contained i n r(C) , and r l r (C) = r l (r (C)) c o n t a i n s r (C) . Thus r l r ( C ) = r ( C ) . S i m i l a r l y , l r l ( C ) = l ( C ) . By (3) of P r o p o s i t i o n 1.1, ( l r ( C ) , r ( C ) ) i s a t o r s i o n theory. I t i s obviously the unique t o r s i o n theory generated by C. S i m i l a r l y , ( 1 ( C ) . , r l (C) ) i s the unique t o r s i o n theory cogenerated by C. Moreover, l r ( C ) = {MJ£ fl,r (C) ]=0] , and 6 r l ( C ) = (H|£l<£) ,M]=0}. a consequence of t h i s observation i s the f o l l o w i n g P r o p o s i t i o n . 1.10. P r o p o s i t i o n . {10, P r o p o s i t i o n 8.1} The s e t of a l l t o r s i o n t h e o r i e s i s a complete l a t t i c e . 1... 11*. P r o p o s i t i o n . L e t (T,F) be a t o r s i o n theory. Then O) C2*£) i s generated by a s i n g l e module, namely, the d i r e c t sum of a l l R/&, where A v a r i e s over dense l e f t i d e a l s . (2) i s cogenerated by a s i n g l e module, namely, the d i r e c t product of a l l R/A, where k v a r i e s over c l o s e d l e f t i d e a l s . (3) (1#D i s a l s o cogenerated by the d i r e c t sum of a l l R/A, where A v a r i e s over c l o s e d l e f t i d e a l s . Proof. (1) Let t=(T,£J and m be the t o r s i o n theory generated by the d i r e c t sum of H/A*s, where A i s dense. C l e a r l y t>a. Suppose t#u, then there i s a l e f t i d e a l A , which i s t-dense but not m-dease. Let ffl<B/A)-B/A, where B#H. Then B i s t-dense and m-closed. But t h i s i s a c o n t r a d i c t i o n , because every t-dense l e f t i d e a l , by the d e f i n i t i o n of 8, i s m-dense. (2) and (3) can be proved i n a s i m i l a r way. 7 Section 2. flings and Modules of Quotients In t h i s s e c t i o n we s h a l l d e s c r i b e the c o n s t r u c t i o n of q u o t i e n t s , which i s the best known g e n e r a l i z a t i o n of the commutative l o c a l i z a t i o n technique. The perfectness of a t o r s i o n theory i s a property shared by a l l commutative l o c a l i z a t i o n s , but not n e c e s s a r i l y retained i n the general case. Let (T,F) be a t o r s i o n theory with corresponding IKF t and corresponding topology D. D i s a d i r e c t e d s e t with i t s p a r t i a l order defined by i n c l u s i o n . For each F i n F, d e f i n e Q t{F) = Q{F) = l i ^ { £ 0 # F J l D i s i n D} as the d i r e c t l i m i t of the a b e l i a n groups £D,Fj* He d e f i n e an B-module s t r u c t u r e of Q(F) i n the f o l l o w i n g way: Let r be i n fi and x i n Q(F). x i s represented by some f i n J.D,F}. Define rx as the element i n Q(F) represented by g i n £ £ D:r J,F ], where g (s)=f (sr) . I t i s easy t o v e r i f y that t h i s m u l t i p l i c a t i o n i s w e l l d e f i n e d , and that Q(F) becomes an H-module with t h i s m u l t i p l i c a t i o n . For M i n H, define Q (M) = Q<M/t(M)). I t i s not d i f f i c u l t to see that Q: M—>M i s a f u n c t o r . 2. 1. D e f i n i t i o n . Q (M) i s c a l l e d the module of quotients , with respect t o the IKF t , of M. 2.2. D e f i n i t i o n , a module E i n M i s c a l l e d t - i n j e c t i v e i f , f o r any M, N i n H such that u i s contained i n M and M/N i s 8 t - t o r s i o n , the n a t u r a l map [M,E]—>£N,EJ i s s u r j e c t i v e . E i s c a l l e d f a i t h f u l l y t - i n j e c t i v e i f , under the same c o n d i t i o n s , i H,E j——>i N, E ] i s one-one and s u r j e c t i v e . -2.3- P r o p o s i t i o n , i 10, P r o p o s i t i o n s 4.1 and 5.1] (A) The f o l l o w i n g c o n d i t i o n s are e q u i v a l e n t : (1) E i s t - i n j e c t i v e . (2) For each t-dense l e f t i d e a l D, £E,E j — » £ D, E j . (3) I{E)/E i s t - t o r s i o n - f r e e . (B) The f o l l o w i n g c o n d i t i o n s are e q u i v a l e n t : (1) E i s f a i t h f u l l y t - i n j e c t i v e . (,2) E i s t - i n j e c t i v e and t - t o r s i o n - f ree. (3) E i s t - t o r s i o n - f r e e , and, f o r each t - t o r s i o n -f r e e F c o n t a i n i n g E, F/E i s t - t o r s i o n - f r e e * 2.4. P r o p o s i t i o n . £12, Theorem 3.7J Q(M) ; i s , up t o isomorphism, the unique f a i t h f u l l y t - i n j e c t i v e module s a t i s f y i n g : (1) H/t(H)>—>Q.(B) , and (2) Q(H)/(M/t (H)) i s t - t o r s i o n . . The map i n (1) assigns m i n H/MM) to f - i n .£B,-fl/t(H>.J such t h a t f(r)=rm. 2.5. P r o p o s i t i o n . (20, P r o p o s i t i o n 0.7] Let F be a t -t o r s i o n - f r e e module with i n j e c t i v e h u l l I . Then Q (F) = p-i ( t ( I / T H ; , where p: I—>I/F i s the n a t u r a l map. E g u i v a l e n t l y , Q (F) = £xjx i s i n I , and £F:xJ i s t-dense}. 9 We are now ready to def i n e a r i n g s t r u c t u r e on Q(B). Let x, y be elements of Q (H) . , Define f i n £B,Q(B) ] by f ( r ) = ry. Since Q{R) i s f a i t h f u l l y t -i n j e c t i v e , f extends to g i n £Q(S) »Q(H) J uniquely. Let xy = g(x)- Then t h i s m u l t i p l i c a t i o n d e f i n e s a r i n g s t r u c t u r e on Q(£), which i s compatible with i t s B-module s t r u c t u r e . I t i s c l e a r that t (B) i s an i d e a l . I t f o l l o w s t h a t B/t (B) i s a r i n g and the n a t u r a l embedding B/t (B) >—>Q(B) i s a r i n g map. The r i n g Q(B) i s c a l l e d the r i n g of quotients (with respect to t) . 2.6. P r o p o s i t i o n . J. 12, Theorem 3.9] The functor Q: H—>M i s l e f t exact. In IKF t i s c a l l e d Noetherian i f i t s a t i f i e s the f o l l o w i n g c o n d i t i o n s : Bhenever A < 1 2 > < . » . i s an ascending chain of l e f t i d e a l s whose union i s t-dense, there i s some n such t h a t a<n> i s t-dense. a module P i s t - p r o j e c t i v e i f f o r any epimorphism H — » N of t o r s i o n - f r e e modules and any map P—>N of B-modules, there i s a dense submodule P f of P and a map P'—>M of modules such th a t the f o l l o w i n g diagram i s commutative. 10 pt> >p H » N 2.7. P r o p o s i t i o n . £ 12, Theorem 4.4 J An IKF t i s Hoethe.ri.an i f and only i f the fu n c t o r Q: M—>H preserves d i r e c t sums. 2.8. P r o p o s i t i o n . £12, Theorem 4.5] Q: M—>M i s r i g h t exact i f and only i f every t-dense l e f t i d e a l i s ^ p r o j e c t i v e . Let A denote the f u l l subcategory i n M of a l l f a i t h f u l l y t - i n j e c t i v e modules. Then Q: M— >A i s a fu n c t o r -I t i s , i n f a c t , the l e f t a d j o i n t of the i n c l u s i o n f u n c t o r A—>H* T h i s means, f o r every M i n H and every A i n A, CQ » a 3 ^ ^  £M,A3 R and the isomorphism i s n a t u r a l i n M and i n A. However, the i n c l u s i o n functor A—>M does not, i n general, preserve coproducts and epimorphisms. .For example, i f A and B are i n A such that B i s a submodule of A and A/B i s i n T, then the i n c l u s i o n B—>A i s an epimorphism i n A, but i s not n e c e s s a r i l y e p i c i n M. Thus the l e f t a d j o i n t a e s s of Q: M—>A i s not too i n f o r m a t i v e . Be turn our a t t e n t i o n t o the category 0 of Q(R)— modules. Every f a i t h f u l l y t - i n j e c t i v e module i s n a t u r a l l y a Q(B)-;mQdule £1.2, C o r o l l a r y 4-23- Thus there i s a f u l l embedding A—>£. The r i n g map B—>Q(B) induces another f u l l embedding Q.—>M. C l e a r l y , the composition f u n c t o r A—>Q—->I i s the n a t u r a l embedding of A i n t o M. I t i s i n t h i s sense 11 that we consider A and Cj as f u l l subcategories of M, with A contained i n 0,. 2 . 9 . P r o p o s i t i o n . The f o l l o w i n g a s s e r t i o n s are e q u i v a l e n t ; (1) Q i s contained i n F. (2) For every t-dease l e f t i d e a l D, Q (S) i (D) = Q (fi) , where i ; B—>Q (fi) i s the n a t u r a l map. (3) = A. (4) Q(B)8 _: M—->£ and Q: M—>0. are n a t u r a l l y e q u i v a l e n t f u n c t o r s . <5) Q: H—>M i s r i g h t exact and preserves d i r e c t sums. (6) Q: M—>p. i s the l e f t a d j o i n t of the f u l l embedding 2—>I, with the ad jugant [ Q (M) ,D — > £ H , . D J R defined by r e s t r i c t i o n . Proof. The equivalence of (1) t o (5) i s proved i n [ 1 2 , Theorem 4 . 3 J . The embedding p.—>H i s n a t u r a l l y e q u i v a l e n t t o i R,_ J. I t i s w e l l known that J. B,_ j : Q—>M i s r i g h t a d j o i n t to Q(fi) ®^_: M—>Cj. Therefore (4) and (6) are e q u i v a l e n t . 2.,10.,, D e f i n i t i o n . An IKE t i s c a l l e d p e r f e c t i f the a s s e r t i o n s i n P r o p o s i t i o n 2 .9 are s a t i s f i e d . , 1 2 Section 3. Examples Five examples are i n c l u d e d i n the present section-..Some c l a s s i c a l r e s u l t s , such as 3.5 (2), f o l l o w n a t u r a l l y from the d i s c u s s i o n . 3.1. D e f i n i t i o n . A t o r s i o n theory i s c a l l e d f a i t h f u l i f 1 i s t o r s i o n - f r e e . , 3 ±2. ..Example. The c l a s s i c a l l e f t . r i n g of q u o t i e n t . „ Let C be a m u l t i p l i c a t i v e l y c l o s e d subset of a , which co n t a i n s the unity 1 but not 0. Then the f a m i l y of l e f t i d e a l s D = {DJtae i n t e r s e c t i o n of D and C i s non-empty} i s a topology i f and only i f the f o l l o w i n g c o n d i t i o n i s s a t i s f i e d : For every c i n C, and every r i n fi, there are c' i n C and r* i n E, such that r , c = c , r . I f t h i s happens, C i s c a l l e d a l e f t Ore s e t , and the IKF defined above i s denoted by /1(C). Let C(fi) be the m u l t i p l i c a t i v e l y closed set of two sided n o n - z e r o - d i v i s o r s of H. Suppose a l s o t h a t C (R) i s l e f t Ore. Denote the IKF /t(C (R)) by c l . Then (1) For every H i n H, c l (Mj i s the s e t of a l l elements m of M which are a n n i h i l a t e d by some c i n C(R). (2) c l i s f a i t h f u l and p e r f e c t . (3) Q(S) i s the l e f t c l a s s i c a l r i n g of q u o t i e n t s of R. That i s t o say: there i s an embedding of r i n g s i : R—>Q(R), such t h a t i (c) i s a u n i t i n Q(R) f o r every c i n C(R) , and 13 that every q i n Q(B) i s equal t o i ( c ) - l i ( r ) f o r some c i n C(8) and some r i n B. 3-3. D e f i n i t i o n . L e t a be i n a . The s i n g u l a r submodule of H i s Z(M) = £m|m i s i n a , the a n n i h i l a t o r of m i s e s s e n t i a l i n B j . a i s c a l l e d s i n g u l a r i f Z(a)=8, and non-singular i f z (a) =0. 3.4. Example- The Goldie t o r s i o n theory. The c l a s s F of a l l aon-^singular modules s a t i s f i e s (2) of P r o p o s i t i o n 1-1. Therefore F i s the c l a s s of t o r s i o n - f r e e modules f o r some t o r s i o n theory. This t o r s i o n theory i s c a l l e d the Goldie t o r s i o n t h e o r y , and the correspondending IKF i s denoted by g o l . Some observations and other c h a r a c t e r i z a t i o n s are; (1) Let H be i n B . .2(H) i s an e s s e n t i a l submodule of go l (H) £34, P r o p o s i t i o n 3.5]. (2) Since Z (fl) i s s i n g u l a r f o r each U i n B, the Goldie t o r s i o n theory i s generated by the c l a s s of a l l s i n g u l a r modules. I t f o l l o w s from t h i s f a c t t h a t D i s the s m a l l e s t topology c o n t a i n i n g the f a m i l y of a l l e s s e n t i a l l e f t i d e a l s . (3) gol ( a)/Z ( a ) = Z(B/Z (a)) £34, P r o p o s i t i o n 3.5J. (4) I t f o l l o w s from (3) that a module a i s g o l - t o r s i o n i f and only i f H/Z(a) i s s i n g u l a r . In p a r t i c u l a r , I(H)./H i s g o l - t o r s i o n f o r every a i n a , where 1(a) i s the i n j e c t i v e h u l l of a . By P r o p o s i t i o n 2.5, Q(a) = I (S/gol (M) ) f o r every a i n a . 14 (5) Suppose the set,, g i g ! _BaazZg£9~diy4^Q3?s of B i s ^ a l e f t Ore s e t . Then the IKF c l i s de f i n e d , and, f o r every c i n C (B), the l e f t i d e a l Rc i s gol-dense. Consequently, c l < g o l * Suppose f u r t h e r t h a t every e s s e n t i a l l e f t i d e a l contains a n o n - z e r o - d i v i s o r . Then c l = g o l . In a semi-prime l e f t Goldie r i n g , every e s s e n t i a l l e f t i d e a l does contain a no n - z e r o - d i v i s o r . 3.5. Example. The maximal l e f t ring_of-_guotieat. The t o r s i o n theory cogenerated by fi, or, e q u i v a l e n t l y , by 1(B), i s c a l l e d the Lambek t o r s i o n theory. By d e f i n i t i o n , i t i s the j o i n of a l l f a i t h f u l t o r s i o n t h e o r i e s i n the l a t t i c e o f IKF's. The IKF corresponding to the Lambek t o r s i o n theory i s denoted by max. The r i n g of quotients Q(B) i s c a l l e d the maximal l e f t r i n g of qu o t i e n t s of B. (1) max < g o l £34, P r o p o s i t i o n 3.9 J . (2) Suppose B i s non-singular. Then B i s g o l - t o r s i o n -f r e e , o r , e q u i v a l e n t l y , g o l i s f a i t h f u l . Consequently, max = q o l . I t f o l l o w s from (4) of Example 3.4 th a t Q ^ x W - Q q0i (B) = 1(B). I t i s a l s o c l e a r t h a t , under t h i s assumption, D ^,ax = D - the f a m i l y of e s s e n t i a l l e f t i d e a l s . In p a r t i c u l a r , i f B i s l e f t Noetherian semi-prime, then max = g o l = c l . Let t be an IKF whose t o r s i o n theory i s cogenerated by an i n f e c t i v e module C. Let H = Cc#c3^ *>e the endomorphism r i n g of C. C can be considered as an B-H bimodule. The 15 endomorphism r i n g B = [ C C j ^ i s c a l l e d the bicommutator of C. Being f a i t h f u l l y t - i n j e c t i v e , C has a n a t u r a l 0_t(B) ^  module s t r u c t u r e such that lc*c~ Qt(f{)~ H. Define a r i n g map k: Q£ (B)—>B by k (g) (cj. = gc, f o r g i n Q (Bj and c i n C. Then k i s an embedding because ker (kjni(B) = i ( A n n R ( C ) ) = i(t(fij) = 0, where i and j are the maps i n d i c a t e d i n the commutative diagram below: fi j >B \ / V / Q t(B) Hhen C i s a f i n i t e l y generated H-module, or when t i s f a i t h f u l , i t i s known tha t k i s a r i n g isomorphism [ 2 1 , P r o p o s i t i o n B j [34, Theorem 8-4 j . In p a r t i c u l a r , we have the f o l l o w i n g r e s u l t : (3j Q M a x{B) B, the bicommutator of 1(H), as r i n g s . 3.6. Example. The t o r s i o n theory at,;a prime i d e a l . Let P be a prime i d e a l of fi. The t o r s i o n theory cogenerated by the fi-module B/P i s c a l l e d the t o r s i o n theory at P. The corresponding IKF i s denoted by Tp. f On the other hand, l e t p: B—>B/P be the n a t u r a l map and C(P)••••= C c l c i s i n B, p (c) i s a n o n - z e r o - d i v i s o r of the r i n g B/P} = { c j f o r every x i n B, cx i s i n P or xc i s i n P i m p l i e s x i s i n P}. Then i t i s easy to v e r i f y that the f a m i l y of l e f t i d e a l s D = (D|for every r i n J , £ D : r j n c ( P ) 16 i s non-empty} of B i s a topology- The IKF corresponding to D i s denoted by /Up . when C (P) i s a l e f t Ore set of B, 0 = { D l D f l C ( P ) i s not empty}. (1) fiip< Xp. Proof. Suppose D i s not tp-dense. There i s a non-zero map f i n £B/D,I (B/P) }. I t f o l l o w s that there i s an x i n B such t h a t 0 # f ( f x 3 ) i n fi/P. Then £D:x]|f (£x3) = 0 i n fi/P. Therefore £D:x]flC(P) i s empty, and D i s not /^-dense. (2) I f every e s s e n t i a l l e f t i d e a l of the r i n g B/P  contains a non-zero-divisor f f then flip = Tp £23, P r o p o s i t i o n 2.2]-Due t o a r e s u l t of Goldie £11, Theorem 3-9 ], t h i s c o n d i t i o n i s s a t i s f i e d by a l e f t Noetherian r i n g . (3) Suppose C(P) i s a l e f t Ore set of B. Then one e a s i l y sees from the c o n s t r u c t i o n of the r i n g s of quo t i e n t s that each c i n C{P) i s a u n i t i n 0.^(8). Thus jU-p i s p e r f e c t by (2) of P r o p o s i t i o n 2.9. 3.7. D e f i n i t i o n . A t o r s i o n theory i s symmetric i f every dense l e f t i d e a l contains a dense i d e a l . An IKF corresponding to a symmetric t o r s i o n theory i s c a l l e d a symmetric k e r n e l f u n c t o r , abbreviated as SKF. Let t be an IKF- The greatest SKF l e s s than or equal t , i f i t e x i s t s , i s denoted by t°. 17 3 . 8 . Example. The s y a a e t r i c , t o r s i o n theory a t a prime i d e a l . Let P be a prime i d e a l of 8, and C(P) be the set defined i n Example 3 . 6 . The f a m i l y D of l e f t i d e a l s c o n t a i n i n g some i d e a l (c) •= BcB, where c i s i n C<P), i s a topology. The corresponding IKF, denoted by Cp , i s symmetric. <1) Since the i d e a l (c) i s ^ - d e n s e f o r every c i n C (P) , Op < jUp < Tp . (2) Suppose every e s s e n t i a l l e f t i d e a l of the r i n g B/P contains a n o n - z e r o - d i v i s o r . Then 0~p = f>p° = [27, Example before P r o p o s i t i o n 12 j . S e c t ion 4. Prime Kernel Functor In t h i s s e c t i o n the notion of prime k e r n e l f u n c t o r (PKF) i s defined. As an immediate consequence of a r e s u l t due to Lambek and M i c h l e r , we show t h a t , t o every prime i d e a l P of a l e f t Noetherian r i n g , there corresponds a PKF Tp as defined i n Example 3 * 6 . He a l s o d e s c r i b e a c o n d i t i o n under which the r i n g of guotients of B with respect to a PKF i s simple A r t i n i a n . 4.1. D e f i n i t i o n . Let t be an IKF. A l e f t i d e a l C i s t -c r i t i c a l i f i t i s maximal among t - c l o s e d proper l e f t i d e a l s . C i s c r i t i c a l i f i t i s t - c r i t i c a l with respect t o some 18 IKF t . In the f o l l o w i n g , the IKF whose t o r s i o n theory i s cogenerated by a module S w i l l be denoted by *VS . i f A i s a l e f t i d e a l , t't/j i s a l s o denoted by Tfi t i f there i s no danger of confusion. Every c r i t i c a l l e f t i d e a l C must be t c - c r i t i c a l . I f C i s a c r i t i c a l l e f t i d e a l , then C must be i r r e d u c i b l e , or, i n other words, R/C i s a uniform module. In g e n e r a l , a module i s uniform i f every non-zero submodule i s e s s e n t i a l . A module H i s indecomposable i f H has only two t r i v i a l d i r e c t summands, namely, 0 and M. A module i s uniform i f and only i f i t s i n j e c t i v e h u l l i s indecomposable., A l e f t i d e a l A i s prime i f , f o r any x and y i n R, xRy i s contained i n A i m p l i e s x i s i n A or y i s i n A. I f A i s a prime l e f t i d e a l , then [A:RJ, the unique biggest i d e a l contained i n A, i s a prime i d e a l . 4 . 3 . D e f i n i t i o n s A module S i s a supporting module of an IKF t i f : ( 1 ) S i s t - t o r s i o n - f r e e , and ( 2 ) f o r any submodule S*#0 of S, S/S« i s t - t o r s i o n . 4 * 4 . D e f i n i t i o n . An IKF t i s c a l l e d a prj. me ..kernel f u n c t o r , abbreviated as PKF, i f there i s a supporting module S of t such that t = ^ 5 . I t f o l l o w s from the d e f i n i t i o n t h a t t i s a PKF i f and only i f t i s cogenerated by R/A f o r some c r i t i c a l l e f t i d e a l 19 A. Let t be a PKF, then there i s a unique (up t o isomorphism) t - i n j e c t i v e supporting module Q of t [12, Theorem 6.4j. For any supporting module S of t , Q i s isomorphic to Qfc(S). . The f o l l o w i n g P r o p o s i t i o n i n d i c a t e s the s i m i l a r i t y between PKF and prime i d e a l . 4.5. P r o p o s i t i o n , f1Q # P r o p o s i t i o n 19.11] Let t be a PKF and t (1) , t ( 2 ) be IKF*s. Then t = the meet of t (1) and t (2) i m p l i e s t = t(1) or t = t (2). 4.6. Pr o p o s i t i o n . £22, Theorem 2.13] Suppose 8 i s l e f t Noetherian. An IKF t i s a PKF i f and only i f t = TFI , the IKF cogenerated by 8/A, f o r some c r i t i c a l prime l e f t i d e a l A. 4.7. P r o p o s i t i o n . Let A be a c r i t i c a l prime l e f t i d e a l and P = £ A: 8 J . Then Tn < Tp . Proof. I t s u f f i c e s to show B/P i s Tfi-torsion-free. Suppose (B/P) = J/P# where J i s an i d e a l c o n t a i n i n g P. Since £"J/P,R/A3 = 0, J i s contained i n A. I t f o l l o w s t h a t J = P and % (B/P) = 0. , 4.8. P r o p o s i t i o n . £22, Theorem 3.9] Suppose B i s l e f t Noetherian and P i s a prime i d e a l . Then 7^, i s a PKF. Proof. Due t o a r e s u l t of G o l d i e , B/P>—> Q^, B/A-, f o r 20 some c r i t i c a l prime l e f t i d e a l s A i such that £A*:B] = P# and the image of R/P i n © f c e/B/A^ i s e s s e n t i a l (see £22, Theorem 3.9 ]) . Therefore I (R/P) = I(B / A ^ ) . This isomorphism i m p l i e s that tp < f o r every i. , On the other hand, by P r o p o s i t i o n 4.7, ?/).^ f o r every i ^ Thus Tp= tfi- f o r every i and i s a PKF. Suppose B i s l e f t Noetherian and t - be a PKF, where A i s a c r i t i c a l prime l e f t i d e a l . Then Q = Qfc(B/A) i s independent of the choice of A, and i t can be proved that P = £A:R] i s c h a r a c t e r i z e d by P = {x i n R|x a n n i h i l a t e s a non-zero submodule of Qj £22, Lemma 2.10].. Consequently, i f t = Tpi f o r some prime i d e a l P», then P' must be £A:BJ. In the f o l l o w i n g P r o p o s i t i o n , we desc r i b e those PKF which are equal t o Hp f o r some prime i d e a l P. 4.9. P r o p o s i t i o n . Suppose B i s l e f t Noetherian and A i s a prime c r i t i c a l l e f t i d e a l . Let P = £ A:B ]. (1) ^ = i f and only i f (A;x]/lC{P) i s empty f o r any x not i n A. (2) HA = i m p l i e s AftC(P) i s empty. Proof. Since n o n - z e r o - d i v i s o r s form a l e f t Ore set i n the l e f t Noetherian prime r i n g B/P, Be*P i s 1p-dense f o r every c i n C(P). I t f o l l o w s t h a t T^ (B/A) = B/A, where B = £x i n Rj£A:x] flC(P) i s not empty}. Thus the c o n d i t i o n that f A : x ] f ) C ( P ) i s empty f o r any x not i n A i s eq u i v a l e n t to 21 th a t "^(R/A) = 0 , the l a t t e r , i n t u r n , i s equivalent t o 7Jj = Tp by P r o p o s i t i o n 4.?« (2) f o l l o w s from (1) by t a k i n g x = 1. In the r e s t of t h i s s e c t i o n , we study the r i n g of quot i e n t s with respect to a PKF. Let t = ^ be a PKF, where Q i s the ( f a i t h f u l l y ) t - i n j e c t i v e supporting module of t . Let E =• I (Q) be the i n j e c t i v e h u l l of Q. . Also denote the endomorphism r i n g f ^ E ^ E ] of E by T, f^Q^Q] by S. Then E becomes a r i g h t T-module and Q a r i g h t S-module. 4.10. P r o p o s i t i o n . With the previous n o t a t i o n s , (1) T i s a l o c a l ring.* (2) The Jacobs on r a d i c a l of T i s J (T) = {.f i n T|Qf = O j . (3) T/J (T) = S as r i n g s . (4) S i s a d i v i s i o n r i n g . (5) Q i s a T-submodule of E. Proof. (1) E i s indecomposable i n j e c t i v e , therefore T i s l o c a l £18 Theorem 25.4]. (2) Suppose Qf = 0. Then f i s not a unit i n T. Hence f i s i n J (T) , as I i s l o c a l . Conversely, suppose Qf # 0. The map f: E—>E induces a monomorphism: 0 # Q/(Ker (f) 0 Q)>—>E. I t f o l l o w s t h a t K e r ( f ) f l Q = 0. Thus Ker(f) — 0, because Q i s e s s e n t i a l i n E. Indecomposability of E i m p l i e s f i s a u n i t , i - e * , f i s not i n J ( T ) . 22 (3) Given f i n T, define $ (f) i n S t o be the unique extension of f: Qf)Q£~l—>Q. Then Q-: T-r->S i s a r i n g map. K e r ( j 9 ) = (f i n T|f i s zero on Qf)Q£-1} = {f i n T|f i s zero on Q} = J (T) . , (4) This i s an immediate consequence of (3). (5) L e t f be i n T. Be have t o show that Qf i s contained i n Q. The map 8 (f) : Q~>Q defined i n (3) extends to g; E—>E by i n j e c t i v i t y . The map f-g i n T i s zero on QDQf-*# hence i s zero on Q. Thus Qf = Qg = Q (f) i s contained i n Q. A module H i s q u a s i - i n j e c t i v e i f , f o r every submodule N of H, every map i n {N,M ] extends to a map i n itJ,Mj. E g u i v a l e n t l y , i s a submodule of i t s i n j e c t i v e h u l l 1(H), as r i g h t £ I (H) ,1 (M) j-modules £7, P r o p o s i t i o n 19.2(c) ]. P r o p o s i t i o n 4.10(5) means p r e c i s e l y t h a t Q i s g u a s i -i n j e c t i v e . 4.11. D e f i n i t i o n , h module M i s c o f a i t h f u l i f the f o l l o w i n g e q uivalent c o n d i t i o n s hold: (1) B>—>&" as H-modules f o r some p o s i t i v e i n t e g e r n, where M** i s the d i r e c t sum of n copies of H. (2) H generates the c l a s s of i n j e c t i v e H-modules. That i s t o say, every i n j e c t i v e module i s a homomorphic image of a d i r e c t sum of copies of M. Using the f a c t that B i s p r o j e c t i v e , the equivalence of 23 the two previous c o n d i t i o n s can be e a s i l y v e r i f i e d - A c o f a i t h f u l module M i s n e c e s s a r i l y f a i t h f u l , i . e . , Anng(M) = 0- I f , i n a d d i t i o n , fl i s t - t o r s i o n - f r e e f o r some IKF t , then t must be f a i t h f u l , because, i n t h i s case, t(B) i s contained i n Ann^(H). 4.12. P r o p o s i t i o n . Suppose t i s a PKF and Q, E, S and T are as i n P r o p o s i t i o n 4.10. Then Q i s c o f a i t h f u l i m p l i e s Q^(B) i s simple A r t i n i a a . Proof. By £7, P r o p o s i t i o n 19.15J, RQ i s i n j e c t i v e and Qs i s f i n i t e l y generated. Therefore E = Q and B = £ET,E^.J i s a matrix r i n g over the d i v i s i o n r i n g T, i . e . , B i s simple A r t i n i a a . But Q t(B) — B as r i n g s [Example 3,5]- Hence Q^(B) i s simple A r t i n i a n . I f M i s t - t o r s i o n - f ree, then t (B) H .= 0. Thus every t -t o r s i o n - f r e e module i s n a t u r a l l y a module oyer the r i n g fi* = B/ t ( B ) . Let F* be the c l a s s of a l l t - t o r s i o n - f r e e modules, considered as B*-aodules. F» i s obviously a c l a s s of t o r s i o n - f r e e B'-modules. Therefore i t d e f i n e s an IKF t ' on the category M« of B ,-modules. I f t i s a PKF on fl with supporting module ^ Q, then t« i s a PKF on M* with supporting module R,Q. Let Q» be the ( f a i t h f u l l y ) t»-injective supporting B*-module of t ' , and E* = I(Q*) i n M*. Also l e t T'« J,. S« = C^QS^Q ' l and B« = f E y , E » T . J -24 4. 13- P r o p o s i t i o n . Suppose t i s a PKF on M and Q i s the t -i n j e c t i v e supporting module of t . I f Q i s c o f a i t h f u l as B * -modules, then Q t(B) i s simple A r t i n i a n , Proof. I t s u f f i c e s to show: (1) Q» = K , Q and (2) Qt ( B ) = Qt 'C f i*) a s r i n g s . This i s the case because, i f these statements are t r u e , Q« w i l l be B»-cofaithful and Qt'(B*) w i l l be simple A r t i n i a n by P r o p o s i t i o n 4.12. (1) i s c l e a r because R»Q i s ( f a i t h f u l l y ) t * - i n j e c t i v e and i s a supporting module of the PKF t * on H«. To see (2), we observe that every t - t o r s i o n - f r e e B-module M i s n a t u r a l l y a t * - t o r s i o n - f r e e B*-module. Hence It^M) = I ( g ' H ) and Qt(M) ~ Qt'(15), as B- and B*-modules. In p a r t i c u l a r , Q t(B) ^ Q t(B/t (fi)) ^ Qt.('Br) as B and B*-modules. Using the f a i t h f u l t - i n j e c t i v i t y of Qt (B), i t i s s t r a i g h t f o r w a r d to v e r i f y t h a t the previous isomorphism Qt«(fi*) = Q t(S) i s i n f a c t a r i n g isomorphism. 4.14. Example. Suppose fi i s l e f t Noetherian and P i s a prime i d e a l . . Let t = l p be the PKF i n P r o p o s i t i o n 4.8. Then P = t(B) i m p l i e s Q t(B) i s simple A r t i n i a n . In p a r t i c u l a r , a l e f t Noetherian prime r i n g has a simple A r t i n i a n l e f t c l a s s i c a l r i n g of qu o t i e n t s . Proof. From the proof of P r o p o s i t i o n 4.8, we see t h a t B/P>—> Q n , where Q i s the t - i n j e c t i v e supporting module of t . I f t(B) = P, then Qt(B) i s simple A r t i n i a n by P r o p o s i t i o n 25 4. 1 3 . I n p a r t i c u l a r , take P = 0 i n a l e f t Noetherian prime r i n g . Then t = max by the d e f i n i t i o n of t , and max = c l f Example 3.5(2) ]- Therefore Q c j (fi) i s simple A r t i n i a n . S e c t i o n 5. Quasi-prime Kernel Functors In t h i s s e c t i o n we study a two sid e d analog of PKF, namely, quasi-prime kern e l f u n c t o r s . The main r e s u l t i s t h a t i n a l e f t Noetherian r i n g every quasi-prime k e r n e l f u n c t o r i s Cp f o r some prime i d e a l P. Let t be an IKF, r e c a l l t h a t t» i s the g r e a t e s t SKF l e s s than or equal to t , provided such a SKF e x i s t s . On the other hand, i f the s e t B* of a l l l e f t i d e a l s c o n t a i n i n g some t-dense i d e a l i s a topology, the corresponding IKF i s then symmetric and i s denoted by t * . Obviously, i f t * e x i s t s , then t° = t * . The f o l l o w i n g P r o p o s i t i o n i m p l i e s , i n p a r t i c u l a r , that t * e x i s t s f o r every IKF t i n a l e f t Noetherian r i n g (see [ 28 ]) . 5.1. P r o p o s i t i o n . Let t be an IKF and I be the s e t of t -dense i d e a l s . I f I has a base of i d e a l s which are f i n i t e l y generated as l e f t i d e a l s , then t * e x i s t s . Proof. He have t o show D* s a t i s f i e s (1) : and (2) of 26 D e f i n i t i o n 1,5. ('I) I f a l e f t i d e a l D c o n t a i n s a t-dense i d e a l I and x i s i n 8, then £D:xJ contains the same i d e a l I . (2) Suppose A i s a l e f t i d e a l c o n t a i n i n g a t-dense i d e a l I , B i s a l e f t i d e a l such that [ B : a | c o n t a i n s a t -dense i d e a l f o r every a i n A. We have to show B contains a t-dense i d e a l . We can assume that I i s f i n i t e l y generated as H a l e f t i d e a l s I = ZL fia-. Suppose I £ i s a t-dense i d e a l *=' 1 1 i s a t -h dense i d e a l . Let J = 21 Ka?. Then obviously J i s contained i n B. J i s an i d e a l because , f o r x i n 8, (Ka k)x = KCJTx^a;) i s contained i n J . J i s t-dense because I / J i s t - t o r s i o n . , Therefore B c o n t a i n s the t-dense i d e a l J . 5.2. D e f i n i t i o n . L e t t be an SKF, an E-bimodule S i s a quasi-support of t i f : (1) S i s t - t o r s i o n - f r e e , and (2) f o r any aon^zero sub-bimodule S' of S, S/St i s t ^ t o r s i o n . 5.3. D e f i n i t i o n . , An SKF t i s a quasi-prime k e r n e l f u n c t o r , abbreviated as QPKF, i f there i s a quasi-support S of t such that ^ * e x i s t s and equals t . The study of IKF and QPKF was i n i t i a t e d by Murdoch and Oystaeyen* I n £28], a non-commutative v e r s i o n of a f f i n e v a r i e t y theory was defined on the spectrum of a l e f t Noetherian prime r i n g 8, where the s t a l k at each P i n SpecB i s Qg, (R). I t i s known that i n a l e f t Noetherian r i n g every 27 Op f o r a prime i d e a l P i s a QPKF £27, Example j . The f o l l o w i n g P r o p o s i t i o n i n d i c a t e s some s i m i l a r i t y between QPKF and prime i d e a l . 5.U- P r o p o s i t i o n . Let t be a QPKF and t ( 1 ) , t(2) be SKF*s-Then t eguals the meet of t {1) and t (2) i m p l i e s t = t (1) or t = t (2) . Proof. Let t = where s i s a guasi-support of t . Then 0 = t(S) = t (1.) (S) f) t (2) (S) . Therefore t(1) (S) = 0 or t(2) (S) -= 0. I f t(1) <S) = 0, then t{1) < T$m Hence t(1) < ^ * = t and t(1) = t . S i m i l a r l y , i f t (2) (S) = 0, then t (2) = t . Denote the set of t - c r i t i c a l i d e a l s (excluding B) by C ( t ) . Due to the f a c t t h a t t-dense i d e a l s are m u l t i p l i c a t i v e l y c l o s e d , C(t) c o n s i s t s only of prime i d e a l s . The f o l l o w i n g P r o p o s i t i o n i s c l e a r . 5.5. P r o p o s i t i o n . L e t B be l e f t Noetherian and s and t be SKF's. I f C(s) = C ( t ) , then s = t . Suppose S i s an B-module and K i s a r i g h t i d e a l of B. Define r (K) = £x i n S| Kx - 0). In ge n e r a l , r (K) i s a submodule. I f S i s an B-bimodule, then r(K) i s a sub-bimodule. Define G(S) = (c i n B| r (cB) = Oj, where S i s an B-module. C l e a r l y G{S) = G ( I ( S ) ) . 28 5.6. P r o p o s i t i o n . Suppose fi i s l e f t Noetherian. An SKF t i s a QPKF i f and only i f t = Op f o r some prime i d e a l P. Proof. I t has been noted that each cTp i s a QPKF, where P i s a prime i d e a l . Let t = fs* be a QPKF with guasi-support S. Then P = fi-G(S) - (x i n S| r(xfi) # Oj i s an i d e a l , thanks t o the f a c t that S i s a guasi-support. Since fi i s l e f t Noetherian, every i d e a l p r o p e r l y c o n t a i n i n g P c o n t a i n s an element of C(P). Consequently, C(<Tp) = ( P j . By P r o p o s i t i o n 5 . 5 , i t remains to show C(t) ••= {Pj. The l a s t c o n d i t i o n i s equivalent to t h a t an i d e a l A i s t-dense i f and only i f the i n t e r s e c t i o n of A and G(S) i s not empty. Suppose c i s i n the i n t e r s e c t i o n of A and G (S) , which i s not empty. Since £fi/ficfi^ I(S) j = 0, Bcfi - i s 7^-dense, hence t-dense. Therefore A i s t-dense. Conversely, suppose A i s t -dense, hence 1$-dense. Then i t f o l l o w s from £fi/A,I (S) J = 0 n that r{A) ••= 0. Let A = |£ Ba t be f i n i t e l y generated as a l e f t i d e a l * Then f)?. r(ajfi) = 0. But each non-zero sub-bimodule of S i s an e s s e n t i a l submodule, hence r(ajtfi) • •= 0 f o r some k. Consequently, a^ i s i n the i n t e r s e c t i o n of A and3 G(S). 5.7. P r o p o s i t i o n . , Suppose fi i s l e f t Noetherian and t i s a PKF, then t * (which does e x i s t ) i s a QPKF. Proof. There i s a c r i t i c a l prime l e f t i d e a l A such t h a t 29 t = *VQ (see P r o p o s i t i o n 4.6). P = £A:B] i s a prime i d e a l and Tfi ^ % (see P r o p o s i t i o n 4.7). By Example 3-8(2), i t s u f f i c e s t o show t h a t Op < - We show aa i d e a l J i s not T--dense i m p l i e s i t i s not -Op-dense... I f J i s not dense, then £B/J,I (R/A) } # 0. I t f o l l o w s t h a t there i s some x i n 8, which i s not i n A, such that Jx i s contained i n A. But A i s a prime l e f t i d e a l , hence J i s contained i n A. Therefore J i s contained i n P and J i s not Op-dense.„ We in t r o d u c e the f o l l o w i n g technique to v study the quasi-supports of a QPKF and t h e i r r i n g s of oimodule endomorphisms. Let K be a commutative subr i n g of R (with the same i d e n t i t y element) contained i n the centre of R. The enveloping algebra of B over K i s , by d e f i n i t i o n , fie = E ^ f i o , where B° i s the opposite r i n g of fi, i . e . , B° = [x<»|x i s i n E} with a« • b° = (a+b) °, aObP - (ba) o, ..H - i s a r i n g w i t h m u l t i p l i c a t i o n defined by (a8b°)(c®d°) = ac8b°do = ac®(db)o. Let BK be the category whose o b j e c t s are B-bimodules M s a t i s f y i n g km = mk f o r any m i n H, k i n K. .The mprphisms of t h i s category are j u s t maps of B-bimodules. Every M i n B.« i s e an B -module v i a (a«b°)m = amb, f o r a and b i n B, and m i n e M. Let the category of B -modules be denoted by H*. Then M» i s n a t u r a l l y e quivalent t o B K. From t h i s point on, we s h a l l f i x K to be the subring of B t h a t i s generated by 1. Thus B K i s the category of a l l B-30 bimodules. Suppose t = T T* i s a QPKF on M with, guasi-support T and S i s another guasi-support of t . Let tj be the IKF on M' (category of B -modules) cogenerated by T, considered as an B -module. 5.8. P r o p o s i t i o n . S i t h the previous n o t a t i o n s : ( 1 ) f o r any B-bimodule H, t (M) < ^(H) < tj> (H) .-.,, (2) 7^ .(S) i s e i t h e r 0 or S. (3) (S) i s e i t h e r 0 or S. Proof. ( 1 ) For any B-submodule N of t ^ M ) , C N # T 2 R = °-Hence f o r any B*-submodule N of 'Ey(H) # £N,T ] ffe =0. This means t h a t tj{IA) , as an B e-module, i s T y - t o r s i o n . Therefore ^(M) < l y * (H). (2) I f ^ y(S) # 0 , then S/v^(S) i s t - t o r s i o n , hence /cy-t o r s i o n . But S/^.(S) i s a l s o t j - t o r s i o n - f r e e . Therefore ^ T(S) = S. (3) can be proved i n a s i m i l a r way. 5.9. P r o p o s i t i o n . B i t h the n o t a t i o n s of P r o p o s i t i o n 5.8: ( 1 ) I f S i s ^ - t o r s i o n - f r e e and t s * e x i s t s , then ^j* i s a QPKF with guasi-support S. (2) I f S i s T T ' - t o r s i o n - f r e e , then ^* i s a PKF on H 1 with supporting module S. , Proof. ( 1 ) tp(S) = 0 means 1j< %. Hence ^ 7 * < Therefore S i s a guasi-support of and VTj^* i s a QPKF. 31 (2) I t f o l l o w s from P r o p o s i t i o n 5.8(1) t h a t , f o r any non-zero B e-submodule S* of S, S/S* i s /2y»-torsion. Hence S i s a supporting module of T j . , Since (T) - 0 by d e f i n i t i o n , the previous argument shows that T i s a supporting module of - Hence ^* i s a PKF. Suppose t = /Zg* i s a QPKF with quasi-support S. Q t(S) i s n a t u r a l l y an B-bimodule because the r i g h t m u l t i p l i c a t i o n on S by an element of B extends uniquely to a map of l e f t R-modules on Q t(S). I t f o l l o w s that every QPKF can be expressed i n the form t = 7^*, where Q i s a ( f a i t h f u l l y ) t -i n j e c t i v e quasi-support.. Let M be an fi-bimodule* The r i n g of bimodule endomorphisms of a i s equal t o fa,aj R« = U. Consider a as a r i g h t 0-module. The r i n g of 0-endomorphisms [M./.ajyOf a w i l l be c a l l e d the quasi-bicommutator of H. , The q u a s i -bicommutator of a i s equal to the bicommutator of the B e-module a. 5.10. P r o p o s i t i o n . Suppose t = fy* i s a QPKF with ( f a i t h f u l l y ) t - i n j e c t i v e quasi-support Q. Let t* = Ttf and Q* = Qt'CQ)- Denote by 00 the B -module map OC: B e—>B defined by OC (a«b<>) = ab. C ) £Q#QJ^e- [QSQ'l / j e and they are d i v i s i o n r i n g s , (2) I f C t - M t ( f i ) ) = t«(B*), and B / t ( B ) > — > Q n , f o r some n, as B e-modules, then the quasi-bicommutators of Q and Q* are both simple A r t i n i a n . 32 Proof. Let 0 = EQ.,Q]geaad 0' = [Q» ,Q» J Re. He f i r s t observe that Q i s a r i g h t U*-submodule of Q': Let f be a map of fi-bimodules i n U'. Since Q f l Q f - 1 i s a non-zero sub-bimodule of Q, the r e s t r i c t i o n of f tp -.QflQf"-1 extends uniquely t o a map $ (f) : Q—>Q of l e f t B-modules Q* # (f) must be a map of B-bimodules, because, f o r any r i n B, the map f f : Q—>Q defined by f r ( x ) = # (f) ( x r ) - { $ (f) <x) ) r i s zero on Q f l Q f - 1 , hence i s zero on Q„ .Secondly, $ ( f ) , being a map of B €-modules, extends uniguely to an B*-map g: Q»—>Q',« map f-g i n U' i s zero on Qf)Qf - 1# hence i s zero on Q». Therefore Qf = Qg = Q #{f) i s contained i n Q. This shows Q i s a IM-submodule of Q*. Now ( 1 ) i s c l e a r because the map Q : D*r—>U defined i n the previous paragraph i s an isomorphism of r i n g s . To see {2), we no t i c e t h a t , under the hypotheses, Q» i s Be/t«(Be)-cofaithful: fi*/t« <Be )—>B/t (B) >—>Q M>—>Q«" .., BY L7#- P r o p o s i t i o n 19.15], Q» i s a f i n i t e dimensional r i g h t vector space over the d i v i s i o n r i n g 0*. Therefore the g u a s i -bicommutator of Q* i s simple A r t i n i a a . By the previous 33 o b s e r v a t i o n , Q i s a subspace of Q*. Therefore the g u a s i -bicommutator of Q = CQ,Q3U = [Q*Q1U. i s a l s o simple A r t i n i a n . 34 CHAPTER I I LOCALIZATION OF GROUP RINGS The c e n t r a l question of t h i s chapter i s the r e l a t i o n between the r i n g s of quotients of the group r i n g RG and those of the group r i n g RH, where H i s a normal subgroup of G. In t h i s area, there are s t i l l unsolved guestions of very simple appearance (see Section 8). The c o n t r o l l e r of an i d e a l i s a f a m i l i a r notion i n t r a d i t i o n a l s t u d i e s of group r i n g s (see, f o r example, £30]). In Section 6 we s h a l l i n v e s t i g a t e some consequences of the exi s t e n c e of a proper c o n t r o l l e r of a topology. In Section 7, the c o n t r o l l e r of a base of a topology i s s t u d i e d . Some of our r e s u l t s , though . developed out of an independent context, do c o i n c i d e with Louden's r e s u l t s £24], F i n a l l y , i n Section 8, we s h a l l i n t e r p r e t two w e l l known r e s u l t s by the th e o r i e s of c o n t r o l l i n g subgroups. Se c t i o n 6. The C o n t r o l l e r Let G be a group and B be a r i n g . By the group r i n g BG we mean the r i n g c o n s i s t i n g of a l l the f i n i t e formal sums £rg with r i n R, g i n G. The a d d i t i o n i s defined componentwise, and the m u l t i p l i c a t i o n i s defined byz ( 21 r„ x) ( X S j y) = SI ( 21 r s ) z . 35 Let H be a subgroup of G. Suppose X i s a l e f t t r a n s v e r s a l f o r fl i n G, i . e . , G i s the d i s j o i n t union of xH, where x i s i n X. We always assume that 1 i s i n X- Then SH i s a s u b r i n g of BG, and BG i s a f r e e r i g h t BH-module generated by X. There i s a map QH: BG—>BH of RH-bimodules, defined by $u ( Sl r„ g) = ,21 r#h. E q u i v a l e n t l y , i f a = 21 xa«, where the 3&i 3 ten h *6X a x*s are i n BH, then 9^ (a) = a^. Moreover, a x ~ By (x~ la) . Suppose I i s a l e f t i d e a l of BG., I t i s c l e a r that (1) &H(I) > I 0 BH. (2) (BG) ( BH{I)) > I > (RG) (I DBH). The f o l l o w i n g P r o p o s i t i o n describes the case where e q u a l i t i e s occur i n (1) and (2). 6.1. P r o p o s i t i o n . With the previous n o t a t i o n s , the f o l l o w i n g a s s e r t i o n s are e q u i v a l e n t : (1) $k CD = I 0 BH. (2) (BG) { fa (I)) = (3) (BG) (I fl BH) = I . m &H (I) * I (5) I = 51 x 6\d). The proof i s easy and hence omitted. We remark t h a t , when H i s normal, X i s a l s o a r i g h t t r a n s v e r s a l f o r H i n G. In t h i s case, (5) can be w r i t t e n as I = SL~ (x fi„(I)x-»)x, where x $^(I)x-* i s a l e f t i d e a l of BH, f o r each x. 36 6.2. D e f i n i t i o n . I f the c o n d i t i o n s of P r o p o s i t i o n 6.1 are s a t i s f i e d , fl i s s a i d to c o n t r o l I. Let F be a f a m i l y of l e f t i d e a l s , H i s s a i d t o c o n t r o l F i f H c o n t r o l s every member of F. Let H < K be subgroups of G such t h a t H c o n t r o l s a l e f t i d e a l I , then K c o n t r o l s I . On the other hand, i f fl and K are subgroups of G c o n t r o l l i n g a l e f t i d e a l X, then the i n t e r s e c t i o n of fl and K c o n t r o l s I . By these observations we can prove a generalized v e r s i o n of [ 2 , Lemma 2.3]; 6.3. P r o p o s i t i o n . Let F be a f a m i l y of l e f t i d e a l s of RG. Let H(F) be the i n t e r s e c t i o n of a l l subgroups of G c o n t r o l l i n g F. Let N (F) be the i n t e r s e c t i o n of a l l normal subgroups of G c o n t r o l l i n g F. Then: (1) and N (F) c o n t r o l F. (2) a subgroup fl c o n t r o l s F i f and only i f fl > H (F) ; a normal subgroup K c o n t r o l s F i f and only i f K > N<F). 6mAm- P r o p o s i t i o n . I f F i s a f a m i l y of l e f t i d e a l s of RG such that Ig i s i n F f o r every I i n F and every g i n G, then W(F) = N(F). Proof. Let fl - 8 (F). I t s u f f i c e s to show gHg-i c o n t r o l s F f o r any g i n G. This a s s e r t i o n f o l l o w s from t h a t <9 -,M) = 9 fe(g-lig)g-1 ^ g 8Ha9)9~ l * g ( i g ) g - 1 = i -37 The normal subgroup N (F) i s c a l l e d the c o n t r o l l e r of F. As an example, l e t B be commutative and F be the s e t of a l l i d e a l s of BG which are r i g h t a n n i h i l a t o r s . A r e s u l t from £2 1 says A *G < N (F) = W (F) < A G, where ^ G = fx i n G|£G?C(x) ] i s f i n i t e ) i s the F.C. Subgroup of G, and JL\ *G i s the t o r s i o n subgroup of ^ G. The f o l l o w i n g i s another example: 6.5. Example. Let fl be a subgroup of G. The augmentation ( l e f t ) i d e a l of H i s the l e f t i d e a l cd^R of BG generated by 1-h, where h i s i n H: COQH = HZ BG(1-h) = 7LZ. x ( A) H) , where con = -VHH. By P r o p o s i t i o n 6.1.(5), i t i s obvious that H c o n t r o l s oo^W. We c l a i m : (1) H = W( O^H) . (2) I f fl i s normal, then H = H( (( W^H)W in=1,2,...J) . To see (1), we assume, on the c o n t r a r y , t h a t there i s some x i n H which i s not i n W = W(4)^.H). Since 1-x i s i n COqR, i t f o l l o w s t h a t 1 - 8u/(l-x) i s i n <50^ .H. This i s a c o n t r a d i c t i o n . To see (2) , i t s u f f i c e s t o show t h a t H c o n t r o l s ( O0^H)n f o r every n=1,2,--.. By i n d u c t i o n on n, ( co^tt)" = Z±Z x ( W H ) n . Therefore H c o n t r o l s ( c*s&i\)n. For the r e s t of t h i s s e c t i o n we s h a l l study b r i e f l y the c o n t r o l l e r of a topology. We remark t h a t , by P r o p o s i t i o n H, (F) = H (F) f o r a topology F. Thus we consider the f o l l o w i n g s i t u a t i o n : fl i s a normal subgroup of G. F i s a topology of l e f t i d e a l s of BG. H c o n t r o l s 38 Define the i n d i c a t o r of ¥ to be K (F) = {x i n G|Ix=I f o r some proper dense l e f t i d e a l 1}. We are i n t e r e s t e d i n the c o n d i t i o n K (F) = G. 6.6- P r o p o s i t i o n . With the previous n o t a t i o n s and assumptions, i f K(F) = G, then G/H i s a t o r s i o n group. Proof. Suppose, on the c o n t r a r y , there i s some x i n G such t h a t x n i s not i n H f o r every n=1,2,.... By assumption, there i s a dense l e f t i d e a l I * BG such t h a t l x < I . Let Y be a l e f t t r a n s v e r s a l of <H,x>, the subgroup generated by H and x, i n G. Then {yx w|y i s i n Y, n = an i n t e g e r j i s a l e f t t r a n s v e r s a l of H i n G-, Since BG(1-x) +1 i s dense (hence c o n t r o l l e d by H) , 1 = $ H ( 1 - X ) i s i n BG (1-x) + I . Thus there e x i s t a i n BG and b i n I such that a (1-x) • b = 1- Applying t o both s i d e s , we have (*) ( Z x ' a ( ) (1-x) • £ x l b i = 1 f o r some a.L i n Bfl, b j i n #/y(I) - BHf)I. I n v o l v i n g only f i n i t e sums, equation (*) can be reduced t o : ( iE x*a.) (1-x) • 2__ x^bi = x f c, 0<k<m, or: (**) (a,+b.)+2: x ' ^ - a ^ + b - ) + x-*M-ai) = x f e where a * = x-*ax. The c o e f f i c i e n t s of x 1 on both s i d e s of (**) ,, being e i t h e r 0 or 1, are i n v a r i a n t under the a c t i o n 39 y " I , { ) ~ x ( ) x - 1 . Thus (**) i s f u r t h e r reduced t o : (***) ( a , t b , ) * X x ' t a ^ -a*_, + b? )•*«••• »-(-a*w ) = x* Comparing the sums of c o e f f i c i e n t s of x on both s i d e s of (***) , »e obtain: Z l b i = ^ x-" b ; x = 1 Thus 1 i s i n I- This i s a c o n t r a d i c t i o n . 6.7. P r o p o s i t i o n . L e t H be a normal subgroup of G which c o n t r o l s a non-zero topology F. In each of the f o l l o w i n g cases, K (F) •= G: (1) There e x i s t s a proper dense i d e a l . (2) H i s c e n t r a l i n G. (3) G i s t o r s i o n . Proof. F i s non-zero means F#{BG}, or, e q u i v a l e n t l y , 1*0. (1) I f I * BG i s a dense i d e a l , then K(F) > {x i n Gl Ix = 1} = G. (2) Every dense l e f t i d e a l i s i n the form I •= 2Z x i * , where I* i s an i d e a l of BH. Thus {x i n GiIx = I j = G f o r every I i n F. (3) Let x be i n G and I be a proper dense l e f t i d e a l . Since x w = 1 f o r some int e g e r n, J = f l ; " I x c i s a dense l e f t i d e a l s a t i s f y i n g Jx = J . 40 Section 7. The Rings Of Quotients.Of_A Group Binq 7.1. D e f i n i t i o n . A normal subgroup H of G weakly c o n t r o l s a topology F of l e f t i d e a l s of BG i f i t c o n t r o l s a base of F. 7.2. P r o p o s i t i o n . ; Suppose F i s a f a m i l y of l e f t i d e a l s of BG, F* i s a fam i l y of l e f t i d e a l s of BH, where H i s a normal subgroup of G. Then the f o l l o w i n g a s s e r t i o n s are e q u i v a l e n t : (1) F i s a topology; H weakly c o n t r o l s F; F* - { 0H(») ID i s i n F j . (2) F* i s a topology; f o r any g i n G and D* i n F*, g-iD«g i s i n F f ; F = {D|D i s a l e f t i d e a l of BG c o n t a i n i n g 51 xD* f o r some D* i n F»j, where X i s a l e f t t r a n s v e r s a l f o r H i n G. (3) F' i s a topology; f o r any x i n X and D» i n F*, x-»D*x i s i n F«; F•= {D|D i s a l e f t i d e a l of RG c o n t a i n i n g 2Z xD« f o r some D" i n F * l , where X i s a l e f t t r a n s v e r s a l **y -f o r H i n G. Proof. To show (2) and (3) are e g u i v a l e n t , we assume (3) and w r i t e g = xh, where x i s i n X and h i s i n H. Then g-iD'g = x-*D*xh = £x-*D*x:h-JJ i s i n To show (1) i m p l i e s (2) « we suppose ( 1 ) holds. I t i s obvious that F equals the s e t of a l l l e f t i d e a l s of BG c o n t a i n i n g some xD», where D* i s i n F«. Let D' = $«(D) be i n F* with D i n F. Then f o r any x i n G, x-tDfx = Qfl ( X - * D X ) = (Dx) i s i n F', because Dx i s i n F. Thus i t 41 remains to show that F* i s a c t u a l l y a topology-Let a» be i n BH and A* = #H (A) be i n F* with A i n F. He show B ,a* < A* f o r some B 1 i n F'.. There i s some B i n F such that Ba« < A- Hence ^(B)a» < B»{k) = A«, where 0^ (B) i s i n F». Next l e t A» and B« be l e f t i d e a l s of BH such that A* i s i n F* and, f o r any a* i n A*, there i s some C\, i n F* s a t i s f y i n g (C^.Ja* < B»- I e have t o show B* i s i n F» , o r , e q u i v a l e n t l y , B = 2HZ xB* i s i n F.By assumption, A - "ZZ xA* i s i n F., I t s u f f i c e s t o show t h a t , f o r any a = S I xa„ i n A, where a*, i s i n A*, there i s D = S I xD», where D* i s i n F*, such that Da < B- By what we have proved so f a r , D' - O xrB»:a-:lx-a i s such a l e f t i d e a l . To show (2) i m p l i e s ( 1 ) , we f i r s t observe that H c o n t r o l s the base (S~xD'|B* i s i n F 1} of F. Suppose A i s i n F and a = ZzZ x a x i s i n BG. He have t o show there i s some B* i n F* such that { S I xB»)a < A. By assumption, A > 2Z xk* f o r some A* i n F'« Then B» = f) x[k* i<-.,c]x-i i s such a l e f t i d e a l - Suppose next that A and B are l e f t i d e a l s of BG such that A i s i n F and iB:a} i s i n F f o r any a i n A. He have to show B i s i n F, o r , e q u i v a l e n t l y , B > 2- xD» f o r some D1 i n *«-x F*. Suppose A > HZ^ xk*, where A* i s i n F*. Then f o r any a* i n A 1, there i s some EK, i n F» such that { 27 xB*.)n* < B-Let D* = ST ( B « , ) a * . Then D« i s i n F» and B > ST xD«-Louden has introduced the notion of an S-qood topology £24]: Let f : B—>S be a map of r i n g s . Let P" be a topology of l e f t i d e a l s of B. Define F t o be the s e t of a l l l e f t 42 i d e a l s D of S such that i r 1 ( D ) i s i n F*. F» i s c a l l e d S-good i f F i s a topology. In the case of f : HH—>BG, where H i s a normal subgroup of G, the F so defined c o i n c i d e s with that described i n P r o p o s i t i o n 7.2(2), Thus the a s s e r t i o n that (2) i m p l i e s (1) i n P r o p o s i t i o n 7.2 c o i n c i d e s with part of [24>, Theorem 3.5 ]. For the remaining part of t h i s s e c t i o n , we suppose the c o n d i t i o n s (1) to (3) of P r o p o s i t i o n 7.2 are s a t i s f i e d . Denote the IKF*s corresponding t o F and F* by t and t», r e s p e c t i v e l y . Let BG®^ = BG« R W_ be the tensor product fu n c t o r from BH-modules to HG-modules. On the other hand, every BG-module may be considered as an BH-module by r e s t r i c t i o n of s c a l a r s . , 7.3. P r o p o s i t i o n . The f o l l o w i n g e q u a l i t i e s hold i n the category of BG-modules: (1) t(M) - t»(M) f o r any BG-module M. (2) t (BG8H*) = BG®t* (M*) f o r any BH-module M». (3) t{BG) = BG8t* (BH) •-. Proof. (1) and (2) are s t r a i g h t f o r w a r d . (3) f o l l o w s from (2) by p u t t i n g M» = BH. 7.4. P r o p o s i t i o n . For any BG-module Q, Q i s f a i t h f u l l y , t -i n j e c t i v e i f and only i f Q i s f a i t h f u l l y t , - i n j e c t i v e . Proof. By P r o p o s i t i o n 7.3 (1), i t s u f f i c e s t o show Q i s 43 t - i n j e c t i v e i f and only i f Q i s t ' - i n j e c t i v e under the assumption t h a t Q i s t o r s i o n - f r e e . Let M* be an BH-module and Q an BG-module. He have £ M VQ3 RH ^ I BG8M* , Q 3 a s a b e l i a n groups., In f a c t , the map f i n £RG®H»,Q]^ corresponding to f« i n X&'+QIRH i s defined by f (g®m) = g <f• (m)) , f o r g i n G and m i n M«. Therefore, i f we have BG8D» < E < BG, where D» i s i n F« and E i s i n F, then we can c o n s t r u c t the f o l l o w i n g diagram C BG , Q ] u- >CBG8D«, Q3 Suppose Q i s t - i n j e c t i v e , then u i s epic f o r any D* i n F *. Therefore u* i s e p i c f o r any D* i n F«. This shows Q i s t • - i n j e c t i v e . Conversely, suppose Q i s t ' - i n j e c t i v e . We have to show uC i i i s e p i c f o r any E i n F. By P r o p o s i t i o n 7.2 (2), E > BGOD' f o r some D* i n F'. Since u 1 i s e p i c , u<2>u<1> i s e p i c . Let f be i n £E,Q]^. Then u<2>f = u<2>u< *>g f o r some g i n £BG,Qj a /., or u< 2>(f - u<*»g) = 0. But Q i s t - t o r s i o n -f r e e . Therefore f - u<* >g = 0, and u<l> i s e p i c . R e c a l l t h a t an IKF i s Noetherian i f and only i f the corresponding l o c a l i z a t i o n f u n c t o r preserves d i r e c t sums. Let the l o c a l i z a t i o n f unctors of t and t» be denoted by Q 44 and Q', r e s p e c t i v e l y . 7 . 5 . P r o p o s i t i o n . (1) For any BG-module M, Q (M) ^ Q« (H) as BH-modules. (2) Q(f) = Q* (f) f o r any map f of BG-modules. Proof. By P r o p o s i t i o n s 7.3 and 7.4, Q{M) i s f a i t h f u l l y •t*.-injective and Q(M)/M* i s t«-torsion, where a * = M/t(M) = M/t' (M>. This proves (1). (2) f o l l o w s from (1). 7.6. P r o p o s i t i o n , t i s Noetherian i f and only i f t ' i s Noetherian. Proof. I f t i s Noetherian, i t f o l l o w s from P r o p o s i t i o n 7 . 2 ( 2 ) that t * i s Noetherian. Conversely, suppose Q* preserves d i r e c t sums. I f { M;) i s a f a m i l y of BG-modules, then © Q(Hc) i s f a i t h f u l l y t ' - i n j e c t i v e by P r o p o s i t i o n 7 . 5 , hence f a i t h f u l l y t - i n j e c t i v e by P r o p o s i t i o n 7.4. Moreover, ® Q(Mj) c o n t a i n s ( ® M;)/t(©B;) as a t»-dense, hence t -dense, submodule. Therefore © Q (M^) .= Q ( 0 H ; ) . . 7.7. P r o p o s i t i o n . For any BH-module M» and any x i n G, Q» (xM») ~ xQ» (M») as BH-modules. Proof. I f N* i s an a r b i t r a r y BH-module and x i s i n G, then xN* i s considered as an BH-module v i a a»xn»=x(x-*atx)n», a» i n BH, n« i n N». Obviously xQMM') i s t 1 - t o r s i o n - f r e e c o n t a i n i n g (xM») / t * (xH») ^ x (M'/t-* f.H*)) as a dense submodule. I t remains t o show xQ*(M*) i s t • - i n j e c t i v e . The f o l l o w i n g commutative diagram, where D» i s i n F*, (vf) (a*) = x-*f ( x a , x _ 1 ) , completes the proof: £BH,xQ« (H».) j ^ — • >£D» , xQ• (H*) j f i A , (RH ,Q«(M») ] w >fx-lD« X,Q» ( H , ) 3 f t H 7.8. P r o p o s i t i o n . Let M' be an BH-module., (1) There i s a unique monomorphism g making the f o l l o w i n g diagram of BH-modules commutative; Q* (M*) > — - q > Q(BG«M«) H« >- — £ - > BG®M* where f(m») •= 1«m* f o r any m* i n fi', and u and u 1 are the n a t u r a l maps. (2) (BG) q {Q* (H«)) ^ BG 9 Q* (H*) i s an e s s e n t i a l BG-module of Q(BG»H«) . (3) I f e i t h e r (i) £G:H J i s f i n i t e or ( i i ) t« i s Noetherian, then (EG) q (Q* ( ft *) ) "= Q (BG8H*) . Proof. (1) This i s an immediate consequence of P r o p o s i t i o n 7.5, because Q(BG®M*) = Q» (BG«M«) i s f a i t h f u l l y t * - i n j e c t i v e . Since f i s mono and g = Q* ( f ) , q i s a l s o mono. (2) Since q i s mono, (HG) g (Q« (M»)) = <5> xq(Q» (M*) ) = ©xQ*(H*) = RG » Q* . This i s an e s s e n t i a l submodule because i t cont a i n s the e s s e n t i a l submodule (RG«H»)/t» (£G«fl() of Q(RG»M»). (3) By P r o p o s i t i o n 7.7, xg (Q* (M*)> i s t * - i n j e c t i v e f o r any x i n G. I f e i t h e r ( i ) or ( i i ) holds, .fiGq (Q* ( 4 ' < ) ) ' = ©xg(Q*(H»)) i s a l s o t»-injective. Hence i t i s a d i r e c t summand o f Q (RG®M*). I t f o l l o w s from (2) that (BG)g(Q* («•.).) = Q(RG8M*) . 7.9. P r o p o s i t i o n . . (1), There i s a unique r i n g embedding q makinq the f o l l o w i n g diagram i n the category of r i n g s commutative: Q* (RH) T u* RH where f ( a * ) = a* f o r any a* i n RH, and u and u» are the n a t u r a l maps. (2) u (RG) g (Q * (RH)) *= RG ® Q* (RH) as r i n g s , where the r i n g s t r u c t u r e of RG ® Q*(8H) i s defined by (x«a*)(y®b') = xySy-ia'yb*. u (RG) g (Q* (RH)) i s a subri n g of Q (RG) . (3) I f e i t h e r ( i ) «.G:H] i s f i n i t e or ( i i ) t ' i s Noetherian, then u (RG) q (Q* (RH) ) = Q (RG) . q > Q (RG) T U f > RG Proof. (1) Replacing M* by RH i n P r o p o s i t i o n 7.8(1) , we 47 obtain an embedding q: Q »(BH)—>Q(RG) of BH-modules s a t i s f y i n g qu» = uf. Using the f a i t h f u l t ' - i n j e c t i v i t y of QJ (BH) and Q (BG) , i t i s easy t o show t h a t q i s a r i n g map. On the other hand, any r i n g map q must be a map of BH-modules. Therefore q i s unique. (2) By P r o p o s i t i o n 7 . 8 ( 2 ) , u(BG)Q'(BH) = (RG). Q* (BH) = BG 8 Q* (RH) i s a submodule of the RG-module Q(RG). The r i n g s t r u c t u r e on BG « Q1(BH) c o i n c i d e s with t h a t of Q (RG) on the t-dense submodule RG- Therefore RG 8 Q' (RH) = u(BG)Q'(RH) as r i n g s and i s a subring of Q(RG) . (3) f o l l o w s from (2) and P r o p o s i t i o n 7.8(3). 7.10. P r o p o s i t i o n . (1) Q i s r i g h t exact i f and only i f Q» i s r i g h t exact. (2) t i s p e r f e c t i f and only i f t» i s p e r f e c t . Proof. (1) By P r o p o s i t i o n 7.5 (2), Q* i s r i g h t exact i m p l i e s Q i s r i g h t exact. Conversely, we show every D* i n F» i s t ' - p r o j e c t i v e , provided every D i n F i s t - p r o j e c t i v e . Let f : H«—»H» and g«: D » — >H« be RH-maps, where M» and N* are t * - t o r s i o n - f r e e , D' being i n F 1 . By t -p r o j e c t i v i t y of 21 xD» = BG 8 D«, there i s E i n F and an RG-map h: E—>BG 8 H* making the f o l l o w i n g diagram of BG-modules commutative: 48 —>BG 8 Df BG8g« v » B G 8 N* By Proposition 7.2(2), we can assume E = 21 xE f f o r some E* in j?'- Define h*: E»—>J3* by h» = ph, where p i s the projection p: BG8M• © xS'—->M». Then h* i s an BH-map and the following diagram i s commutative; BG 8 M». — BG8f • E» >-h« •> D< M» g* - » N 1 Therefore D• i s t»-projective. (2) follows from (1). and Proposition 7.6, Section 8. The Maximal Ring Of Quotients ftnd The C l a s s i c a l Binq Of Quotients Let G be a group and fl be a normal subgroup. I t i s s t i l l unknown what condition on G and H i s equivalent to that Q J » A X ( B G ) BG ® Q M A X(BH) , and, i f the c l a s s i c a l rings of guotients e x i s t , what condition i s equivalent to that Q c l ( B G ) = BG ® Q i t (BH). Be s h a l l display some known r e s u l t s in the context of Section 7., 49 8.1. Pro p o s i t i o n . £25, C o r o l l a r y 12] Let H be a normal subgroup of G. Then: (1) Q m A y ( f i H ) i s uniquely embedded i n t o Q w 4 y(BG) as a subring such that the f o l l o w i n g diagram i s commutative: > Q BH M A X T U (BG); BG where u«, u and f are the n a t u r a l maps. Moreover, u(BG]q (Q m K l l (BH)) ~ BG 0 Q^yCBH) and i s a subring of Q *^(BG) • (2) I f £G:HJ i s f i n i t e , then BG ® Q ^  (BH) = Q ^^(BG) v i a the previous embedding. Proof. (1) Put F* = the Lambek topology on BH and t» = max^ i n P r o p o s i t i o n 7.2- F» i s G-invariant £24, p.528 ]. By P r o p o s i t i o n 7.2(2), we o b t a i n a topology F of BG, with corresponding IKF t . By P r o p o s i t i o n 7.9, Q r t A X (BH) i s embedded i n t o Qt(BG) as a su b r i n g and BG ® Q m M t (BH) i s a subring of Q t(BG). But F i s contained i n the Lambek topology on BG £9, Theorem 1 j . Therefore, by P r o p o s i t i o n 2.5, Q*(BG) i s a subring of Q m A x (BG). Thus the embedding g i s obtained. The commutativity of the diagram i s obvious. To see that q i s unique, we observe t h a t t i e f o l l o w i n g diagram i s commutative: 50 BG ® Qw*x(SH) BG«g > QmAM (RG) I RG and that (RG 8 Q m M (BH))/RG i s t-torsion, hence raax^-to r s i o n . (2) I f [G:Hj i s f i n i t e , then i t follows from Proposition 7.9(3) that BG 8 Q (BH) = Q t(BG). By [ 25, Theorem 5 and Lemma 10j, F = Lambek topology of BG. Therefore Qt(BG) •= Q m M C(RG). For further r e s u l t s concerning the maximal ring of quotients, one may consult [25, C o r o l l a r i e s 13, 14]. Be now turn to the c l a s s i c a l r i n g of quotients. The following r e s u l t s are e s s e n t i a l l y due to Herstein and Small: 8.2. Proposition, i 161 Let B be a commutative r i n g , G be a group and J G be i t s F.C. Subgroup. (1) I f G/AG i s l o c a l l y f i n i t e , then C (BG) i s a l e f t Ore set (see Example 3.2 and Section 9 for the d e f i n i t i o n s ) . Hence Q c\ (BG) e x i s t s . (2) I f H i s a normal subgroup of G, contained i n 4 G.,. such that G/H i s l o c a l l y f i n i t e , then, f o r any d i n C(BG), there are d* in C(BH) and a in BG such that ad = d*. (3) If H i s a normal abelian subgroup of G such that G/H i s l o c a l l y f i n i t e , then, for any d i n C(RG), there are d* i n C(BH) and a i n BG such that ad = d*. 51 Be remark t h a t i f G/4 G i s l o c a l l y f i n i t e , then f o r any subgroup H, H/4 B i s a l s o l o c a l l y f i n i t e , hence C(fiH) i s a l s o l e f t Ore. The f o l l o w i n g r e s u l t on c l a s s i c a l r i n g of q u o t i e n t s i s analogous to P r o p o s i t i o n 8 . 1 . 8 . 3 . P r o p o s i t i o n , suppose fi i s a r i n g , and both C (£G) and C(BH) are l e f t Ore, where H i s a normal subgroup of G. (1) Q c^ (BH) i s uniquely embedded i n t o Q ^ (BG) as a s u b r i n g such that the f o l l o w i n g diagram i s commutative: Q 4 (BH) g > Q c l (BG) j u BH f > BG where u», u and f are the n a t u r a l maps. Furthermore, u (BG)q (Q c t (BH)) " RG 6 .Q cl (BH) and i s a subring of Q c l (fiG) . (2) Suppose f u r t h e r t h a t B i s commutative, and e i t h e r ( i ) H < A G, G/H i s l o c a l l y f i n i t e , or ( i i ) H i s abelian> G/H i s l o c a l l y f i n i t e . Then fiG ® Q ci(»H) = Q £ j (BG) v i a the previous embedding. Proof. F o l l o w i n g the proof of P r o p o s i t i o n 8 . 1 ( 1 ) , we l e t t» = cl£H to o b t a i n an IKF t on BG. Since C (BH) i s contained i n C(BG), t < c l . Therefore Q ^(RH) < BG ® Q ci (BH) < Q t(BG) < Q ^ (fiG) , and the other a s s e r t i o n s f o l l o w as i n the proof of P r o p o s i t i o n 8 . 1 (1 ) -We remark t h a t , i n f a c t , fiG ® Q c l (BH) = Q t(BG) because 52 c l M i s Noetherian. (2) Under the assumptions, t = c l R ( r . fay P r o p o s i t i o n 8.2(2) and ( 3 ) . Therefore EG ® Qtl (BH) = QC(BG) = Q c j (BG) . 5 3 CHAPTEB I I I GLOBAL DIMENSION OF_GBO0'P BINGS The aim of t h i s chapter i s to i n v e s t i g a t e the ( l e f t ) g l o b a l dimension of the group r i n g BG, where e i t h e r B i s commutative or G i s a l o c a l l y M-group. In p a r t i c u l a r , we s h a l l show t h a t , under s p e c i a l c o n d i t i o n s , the f i n i t e n e s s of l . g l . d . B imposes a f i n i t e upper bound f o r l.gl.d.BG. H i s t o r i c a l l y , our r e s u l t may be considered as a c o n t i n u a t i o n of H i l b e r t * s Syzygy Theorem; i f G i s a f i n i t e l y generated t o r s i o n free a b e l i a n group, then l.gl.d.BG = l . g l . d . B • rank (G). Along t h i s l i n e of approach, Balcerzyk has proved that gl.d.ZG = n + 2, where G i s a t o r s i o n f r e e a b e l i a n group, not f i n i t e l y generated, of rank n, and Z i s the r i n g of i n t e g e r s £3, Theorem J . Section 9. Background I n t h i s s e c t i o n we s h a l l d i s p l a y some w e l l known f a c t s from group theory and homological algebra. These r e s u l t s can be found i n £4], £6J, £15], £17], £26], £33] and £35]. A normal s e r i e s of a group G i s a s e r i e s (*) : 1 = G0< G, < < 6n = G of subgroups of G, such that G i-t i s normal i n Gt-, 1=1,2,...,n. The groups G;/G{-/ , i=1,...,n, are c a l l e d the f a c t o r s of the normal s e r i e s (*). I f each G; i s normal i n G, (*) i s c a l l e d an i n v a r i a n t s e r i e s . I f (*) i s i n v a r i a n t and each Gi/Gt - i i s contained i n the centre of G / G , then (*) i s c a l l e d a c e n t r a l s e r i e s . Normal s e r i e s , i n v a r i a n t s e r i e s and c e n t r a l s e r i e s are a l l f i n i t e by d e f i n i t i o n . The l e n g t h of (*) i s n, the number of i t s f a c t o r s . A group G i s n i l p o t e n t i f i t has a c e n t r a l s e r i e s . In t h i s case, the upper c e n t r a l s e r i e s and the lower c e n t r a l s e r i e s o f G are both f i n i t e and are of equal l e n g t h . This common lengt h i s c a l l e d the c l a s s of the n i l p o t e n t group G. A group i s s o l v a b l e i f i t has a normal s e r i e s w i t h a b e l i a n f a c t o r s . In t h i s case, the derived s e r i e s of G i s f i n i t e . A l l n i l p o t e n t groups are s o l v a b l e . A group G i s c a l l e d supersolvable i f i t has an i n v a r i a n t s e r i e s with c y c l i c f a c t o r s . G i s p o l y c y c l i c i f i t has a normal s e r i e s with c y c l i c f a c t o r s . F i n a l l y , G i s an M—group i f i t has a normal s e r i e s with f i n i t e or c y c l i c f a c t o r s . Obviously, every supersolvable group i s p o l y c y c l i c , and every p o l y c y c l i c group i s an M-group. A group G i s s a i d to have maximal c o n d i t i o n i f every subgroup i s f i n i t e l y generated. E q u i v a l e n t l y , G has maximal c o n d i t i o n i f every s e t of subgroups has a maximal element. H—groups have maximal c o n d i t i o n £33, 7. 14]. Therefore, G i s p o l y c y c l i c i f and only i f G i s s o l v a b l e and has maximal c o n d i t i o n . A group G i s l o c a l l y f i n i t e i f every f i n i t e l y generated 55 subgroup of G i s f i n i t e - Other l o c a l p r o p e r t i e s are defined i n a s i m i l a r way. A group G i s n i l p o t e n t by f i n i t e i f G has a normal n i l p o t e n t subgroup of f i n i t e index. Other b y - f i n i t e - p r o p e r t i e s are defined i n a s i m i l a r way. i t f o l l o w s from [33, 7.1.10] that M—groups, supersolvable by f i n i t e groups and p o l y c y c l i c by f i n i t e groups are equ i v a l e n t terms. Suppose G i s f i n i t e l y generated n i l p o t e n t , , with lower c e n t r a l s e r i e s 1=Z°SZ*<.,,$Zn=G. Then, due to the c e n t r a l c o n d i t i o n , each f a c t o r Z^/Z^-i i s f i n i t e l y generated [15, Lemma 1 . 7 ] and a b e l i a n . Thus a f i n i t e l y generated n i l p o t e n t group i s supersolvable* Consequently, l o c a l n i l p o t e n c y , l o c a l s u p e r s o l v a b i l i t y and l o c a l p o l y c y c l i c a l i t y are equi v a l e n t t o one another. The preceeding d i s c u s s i o n i s summarized i n the f o l l o w i n g diagram a b e l i a n f i n i t e l y gene-rated n i l p o t e n t s u p e r s o l v a b l e li p o l y c y c l i c p o l y c y c l i c ^ ^ s u p e r s o l v a b l e by f i n i t e M-group by f i n i t e n i l p o t e n t by f i n i t e n i l p o t e n t li l o c a l l y n i l p o t e n t % l o c a l l y p o l y c y c l i c l o c a l l y supersolvable l o c a l l y B-group f i n i t e 4 l o c a l l y f i n i t e 5 6 a normal s e r i e s with f i n i t e or c y c l i c f a c t o r s i s c a l l e d an M-series. a normal s e r i e s with only c y c l i c f a c t o r s i s c a l l e d a p o l y c y c l i c s e r i e s . Let G be a group, then any two M-series, provided they e x i s t , have egual number of i n f i n i t e f a c t o r s 133, 7.1.5 ]. , 9.1. D e f i n i t i o n . The H i r s h number h(G) of an M-group G i s the number of i n f i n i t e f a c t o r s of any M-series., The H i r s h number of a l o c a l l y M-group i s the supremum of the H i r s h numbers of i t s f i n i t e l y generated subgroups* We now tu r n t o homological a l g e b r a . Let S be a r i n g , B an S-module. The n-th r i g h t derived f u n c t o r of the f u n c t o r Homs(_,B) from the category of S-modules to the category of ab e l i a n groups i s denoted by Ext£(_,B). Extg(a,B) i s a b i f u n c t o r i n the S-modules a and B. 9.2. D e f i n i t i o n . The p r o j e c t i v e dimension p. d.^ a (or p.d . a , i f S i s understood) of an S-module a i s the s m a l l e s t i n t e g e r n>0 f o r which Ext"^-* (A,^.) = 0, provided such an i n t e g e r e x i s t s . I f there i s no such i n t e g e r n, then p.d. 5 a i s i n f i n i t e . The l e f t g l o b a l dimension 1 .gl.d.S of the r i n g S i s the supremum of p . d . s a , where a v a r i e s over a l l S-modules. I t i s w e l l known that the f o l l o w i n g c o n d i t i o n s are eq u i v a l e n t : (1) p.d . sa = n. (2) the leng t h of the s h o r t e s t 57 p r o j e c t i v e r e s o l u t i o n of A i s n. (3) n i s the l a r g e s t i n t e g e r f o r which Extg(A,_)#Q. 9.3. Proposit i o n . £18, S h i f t i n g Theorem, p.47 ] Suppose A > — > P — » B i s a short exact sequence of S-modules and p i s p r o j e c t i v e . Then Ext{? (A,_) ~ Ext " | M B > _ ) , n=1,2,*,..( Conseguently, p.d.A=0 i f and only i f p.d.B=0 or 1; p-d.A=n i f and only i f p.d.B=n*1, n=1,2,...; and p.d.A i s i n f i n i t e i f and only i f p.d.B i s i n f i n i t e . 9.4. D e f i n i t i o n . Let £ be a r i n g , and G a group. The cohomological dimension c*d.^G of G with respect to R i s p.d.^B, where S i s considered as an BG-module with t r i v i a l G-action. 9.5. P r o p o s i t i o n . Let B be a r i n g , and fi be a subgroup of a group G. Then (1) f o r any BG-module M, p . d . ^ f i < p.d.^M; (2) l . g l . d . f i H < l.gl.d.RG. Proof. By £17, P r o p o s i t i o n 12.3], Ext^(H,Hom f l H{BG,B)) = Extft W{M,B) f o r any HH-module B. Hence <1) f o l l o w s . Let A, B be a r b i t r a r y BH-modules. H-BG^/^K i s an BG-module c o n t a i n i n g A as an BH-direct sumraand. Applying the isomorphism i n the preceeding paragraph, we see th a t E x t ^ ( _ , _ ) = 0 i m p l i e s Ext / J <(A,B)=0. Heace (2) holds. 58 9.6. P r o p o s i t i o n . Let B be a r i n g , and H be a subgroup of a group G. Consider the f o l l o w i n g c o n d i t i o n s ; (1) For any BG-module M, p.d.^M = 0 i m p l i e s p.d.^H < n. (2) For any BG-module H, p.d.^H < p.d.^HM • a. (3) l.gl.d.BG < l.gl.d.BH + n. Con d i t i o n s (1) and (2) are e q u i v a l e n t ; and (2) i m p l i e s (3). Proof. I t i s c l e a r t h a t (2) i m p l i e s (1) and (3). He have t o show (1) i m p l i e s (2). Suppose (2) has been proved f o r a l l BG-modules M* with p.d.^M*<k. Let p.d.^.H=k. Construct a short exact sequence N > — > P — » H of BG-modules, where P i s BG-, hence BH-, p r o j e c t i v e . By the S h i f t i n g Theorem, (2) holds f o r M. Thus (2) i s v a l i d f o r a l l BG-modules of f i n i t e p r o j e c t i v e dimension^Suppose p.d.^M i s i n f i n i t e . Then p.d.^H must be i n f i n i t e , otherwise, by repeatedly a p p l i c a t i o n of the S h i f t i n g Theorem, there i s an BG-module which i s BH-p r o j e c t i v e but of i n f i n i t e BG-projective dimension. This c o n t r a d i c t s {1). 9.7. P r o p o s i t i o n . Let B be a r i n g , and G be a f i n i t e c y c l i c group of order n. I f n i s not a u n i t i n 8, then c.d.^G i s i n f i n i t e . Proof. I t i s w e l l known that Ext 2*+ 2(B,B)=B/nB, f o r a l l 59 n>0 £14, pp.39-40j. I f n i s not a unit i n E, then E x t 2 « * 2 { B , f i ) # 0 , f o r n>0. Therefore c.d.^G i s i n f i n i t e . . S ection 10. The Commutative Case Let B be a commutative r i n g , and G be a group. We s h a l l prove that gl.d.BG < g l . d . f i + c.d.^G. E being commutative, every l e f t BG-module M has a n a t u r a l r i g h t BG-module s t r u c t u r e given by mg=g-*m, f o r every element g of G, and every element m of H. I n t h i s way, a map of l e f t BG-modules i s n a t u r a l l y a map of r i g h t BG-modules. Therefore 1.gl.d.BG = r.gl.d.BG. This common value of the g l o b a l dimension of EG on e i t h e r s i d e i s denoted by gl.d.BG. An BG-module k i s c a l l e d weakly p r o j e c t i v e i f the BG-map f: .BG&^A—>A, defined by f{g«a)=ga, s p l i t s . Obviously, i f A i s p r o j e c t i v e , then i t i s weakly p r o j e c t i v e . 10.1. P r o p o s i t i o n . £31. P r o p o s i t i o n 2.3] Let B be a r i n g (not n e c e s s a r i l y commutative) and G be a group. An BG-module A i s p r o j e c t i v e i f and only i f i t i s B - p r o j e c t i v e and weakly p r o j e c t i ve. Proof. Suppose A i s B - p r o j e c t i v e , then BG®^A i s BG-p r o j e c t i v e . Suppose f u r t h e r t h a t A i s weakly p r o j e c t i v e , 60 then & i s isomorphic to an .BG-direct summand of BG« Ra. Hence a i s BG-projective. The converse i s c l e a r . Suppose B i s commutative and a, B are BG-modules. Then a%B has an BG-module s t r u c t u r e given by the diagonal a c t i o n of G: g (a»b)= (ga)8 (gb), f o r elements g of G, a of A, and b of B. I f A i s weakly p r o j e c t i v e , then the BG-module A8 RB with diagonal a c t i o n of G i s a l s o weakly p r o j e c t i v e [.&,. P r o p o s i t i o n 8.5]. 10.2 P r o p o s i t i o n . Let B be a commutative r i n g , and G be a group. (1) For any BG-module M, p.d.RH < p.d. < p. d.^H • c. d.^G. (2) g l . d . f i < gl.d.BG < g l - d . f i • c.d.^G. Proof. I t s u f f i c e s to prove that p.d.^H = 0 i m p l i e s p . d . ^ f l < n i n the case that c.d.^G = n i s f i n i t e , where H i s an BG-module. The r e s t of the a s s e r t i o n f o l l o w s from P r o p o s i t i o n s 9.5 and 9.6. Since c.d.^G = n, there i s an BG-projective r e s o l u t i o n .p. of B of l e n g t h n. By B - p r o j e c t i v i t y of H, the complex .P.SflM—>B8gM—>0 i s a c y c l i c . Equipped with the diagonal a c t i o n of G, t h i s complex becomes an a c y c l i c complex of BG-modules, while B8^H i s isomorphic to M as BG-modules. Each term i n the complex .P.8^H i s B - p r o j e c t i v e and weakly p r o j e c t i v e , hence i s BG-projective. Therefore .P.«RM i s an 61 BG-projective r e s o l u t i o n of M of length n. Thus p.d.^M < n. Section 11. The F i n i t e Case _And The Case Of H-groups In t h i s s e c t i o n , we s h a l l i n v e s t i g a t e l.gl.d.BG, where B i s an a r b i t r a r y r i n g and G i s a f i n i t e group or an M-group. As a s p e c i a l case of H-groups, we s h a l l use a r e s u l t of Gruenberg ( P r o p o s i t i o n 11.7) to get a stronger r e s u l t i n the case th a t G i s a f i n i t e l y generated n i l p o t e n t group. The f o l l o w i n g P r o p o s i t i o n takes care of the f i n i t e case. 11.1. P r o p o s i t i o n . Suppose B i s a r i n g and G i s a group. (1) I f H i s a normal subgroup of G of f i n i t e index n, and n i s a u n i t i n B, then l.gl.d.BG = 1.gl.d.BH. (2) I f G i s f i n i t e of order n, where n i s not a u n i t i n B, then l.gl.d.BG i s i n f i n i t e . Proof. (1) The argument of £ 19, Appendix 2, Lemma 3] can e a s i l y be modified t o show that every BG-module which i s BH-projective i s a l s o BG-projective. The r e s u l t f o l l o w s from P r o p o s i t i o n s 9.5 and 9.6. (2) As n i s not a unit i n 8, there i s a prime f a c t o r p of n such t h a t p i s not a u n i t i n B. By Cauchy's Theorem £32, Theorem 5.2j, G has a c y c l i c subgroup H of order p. 6 2 According t o P r o p o s i t i o n 9.7, l.gl.d.BH, i s i n f i n i t e . Therefore 1.gl.d.BG i s i n f i n i t e . He now tu r n t o the case where G i s an M-group. By £32, 7.1.10], an M-group G has a normal s e r i e s 1=Ge<G, <-..<GWfi =G, where n=h (G) , such t h a t G/Gw i s f i n i t e and a l l the other f a c t o r s are i n f i n i t e c y c l i c . Such a normal s e r i e s w i l l be c a l l e d a standard H-series. Suppose H i s a normal subgroup of G such that G/H i s i n f i n i t e . Then obviously BG i s isomorphic, as r i n g s , to the t skew group r i n g BH {. x,x _* J, where the m u l t i p l i c a t i o n r s defined by xa=(a*)x, a*=xa.x~1--. Thus the f o l l o w i n g P r o p o s i t i o n i s j u s t another v e r s i o n of £8, Lemma 23]. 11.2. P r o p o s i t i o n . Let B be a r i n g and H be a normal subgroup of G such t h a t G/H i s i n f i n i t e , c y c l i c . Then 1.gl.d.BG < l.gl.d.BH + 1 . 11 . 3 . .Proposition. Let G be an M-group with standard M-s e r i e s 1=G0<. .-<G n t l =G. Let B be a r i n g . Suppose the order of the f i n i t e group G/G„ i s a u n i t i n B. Then l^gl.d.BG < 1.g 1.d. B • n = l . g l . d . B • h (G). The preceeding P r o p o s i t i o n i s a d i r e c t conseguence of P r o p o s i t i o n s 11.1.(1) and 11.2. As a c o r o l l a r y , we have the f o l l o w i n g group-theorectic r e s u l t . 63 11.4. P r o p o s i t i o n . Let G be an M-group with standard M-s e r i e s 1-G0<. . .<G =G such t h a t the order of G/G„ i s m. I f x i s a t o r s i o n element of G of order k, then every prime f a c t o r o f k i s a prime f a c t o r of m., Proof. Deny the conc l u s i o n . Then there i s a c y c l i c subgroup fl of prime order p such th a t p i s not a f a c t o r of m. Let B be the l o c a l i z a t i o n of the r i n g of i n t e g e r s a t the m u l t i p l i c a t i v e s et {1, m, m a,...j. Then m i s a unit i n B, while p i s not. By P r o p o s i t i o n 11.3, 1. g l . d . BG i s f i n i t e . However, by P r o p o s i t i o n s 11.1 and 9.5, 1. gl.d.BG i s i n f i n i t e . This i s a c o n t r a d i c t i o n . He s h a l l now prove a stronger v e r s i o n of P r o p o s i t i o n 11.3, supposing G i s f i n i t e l y generated n i l p o t e n t . The key f a c t t h a t makes the d i f f e r e n c e i s the f o l l o w i n g r e s u l t : Let G be a group, not n e c e s s a r i l y n i l p o t e n t , with upper c e n t r a l s e r i e s l=Z0<Zt <-.., , Then £Z n*i /Z„,Z( j , the set of homomorphisms of the p a i r of a b e l i a n groups Zn+//Z„ and Z,, n=1,2,-.., separates p o i n t s . I n other words, f o r any Z„x#1 i n Znti /Z„, there i s f i n .[Z M * i /Zn,Z, ] such t h a t f ( Z n x ) f 1 i n Z, £35, p.6]. 11.5. D e f i n i t i o n . Let G be a group, and B be a r i n g . G i s fi-t o r s i o n r f r e e i f the order of every t o r s i o n element of G i s a u n i t i n B-64 11.6. P r o p o s i t i o n . Let G be a n i l p o t e n t group with upper c e n t r a l s e r i e s V=Z0<Z|£.-.:'£Zn=G. (1) For k=1,.,.,n, i f Zk/Z«., has a t o r s i o n element of prime order p, then Z, has a t o r s i o n element of order p. Consequently, f o r any r i n g 8, Z , i s fi-torsion-free i m p l i e s each Zk/Zfc., i s 8 - t o r s i o n - f r e e . (2) I f G i s f i n i t e l y , generated t o r s i o n - f r e e , then G has a p o l y c y c l i c s e r i e s with only i n f i n i t e c y c l i c f a c t o r s . (3) I f G i s f i n i t e l y generated, then the set T of t o r s i o n elements of G i s a f i n i t e c h a r a c t e r i s t i c subgroup of G. (4) . I f G i s f i n i t e l y generated, then G has an H-series 1=G,<Gi<..,<GW=G such that G ( i s f i n i t e and the other f a c t o r s are i n f i n i t e c y c l i c . Proof. (1) f o l l o w s from t h a t [Zu/Z*., , Z i ] separates p o i n t s . (2) Since G has maximal c o n d i t i o n , each Zk/Zk., i s f i n i t e l y generated a b e l i a n * These f a c t o r s are a l s o t o r s i o n -f r e e by (1). Hence the upper c e n t r a l s e r i e s can be r e f i n e d to a p o l y c y c l i c s e r i e s with only i n f i n i t e c y c l i c f a c t o r s . (3) The set of t o r s i o n elements of a n i l p o t e n t group i s always a c h a r a c t e r i s t i c subgroup 133, 6.4.13 J. Under the curre n t assumptions, T i s a f i n i t e l y generated, t o r s i o n , n i l p o t e n t group, hence i s f i n i t e . (4) L et Gi=T- Then G/G, i s t o r s i o n - f r e e . The concl u s i o n f o l l o w s from (3) and (2). 11.7. P r o p o s i t i o n . (Generalized v e r s i o n of [14, Theorem 5, P-149J) Let G be a f i n i t e l y generated n i l p o t e n t group, and B be a r i n g . (1) I f G i s not B - t o r s i o n - f r e e , then c.d. RG i s i n f i n i t e . (2) I f G i s B - t o r s i o n - f r e e , then c.d.RG = h (G). 11.8. P r o p o s i t i o n . Let G be a f i n i t e l y generated n i l p o t e n t group, and fl be a r i n g . (1) l.gl.d.BG i s f i n i t e i f and only i f G i s B - t o r s i o n -f r e e and l . g l . d . B i s f i n i t e . (2) I f G i s B - t o r s i o n - f r e e , then c^d.^G < l.gl.d.BG .< l . g l . d . B • c.d. RG. Proof. (2) Since G i s f i n i t e l y generated n i l p o t e n t , i t has f i n i t e H i r s h number h(G) and an M-series 1=Ga<G/<...<Gjt(5)+, =G with G i f i n i t e and the other f a c t o r s i n f i n i t e c y c l i c . By P r o p o s i t i o n s 11.1 and 11.2, l.gl.d.BG < l . g l . d . B • h(G) •= l.gl.d.B + c.d.^G. A l s o c.d.RG < l.gl.d.BG by d e f i n i t i o n . , (1) I f G i s B - t o r s i o n - f ree, then (2) shows t h a t l.glwd.BG i s f i n i t e . I f G i s not B - t o r s i o n - f r e e , then G has a f i n i t e c y c l i c subgroup H of order not a u n i t i n B. Since l.gl.d.BH i s i n f i n i t e , so i s l.gl.d.BG. 66 Section 12. Osofsky's Theorem And G e n e r a l i z a t i o n To L o c a l l y  M-Groups We s h a l l use the f o l l o w i n g theorem of Osofsky to g e n e r a l i z e the r e s u l t s obtained i n the preceeding s e c t i o n . 1 2 . 1 . P r o p o s i t i o n . £Qsofsky's Theorem, 2 9 , Theorem 2 .44 and P r o p o s i t i o n 2 . 3 8 J Let S be a r i n g . Let { A;3j be a d i r e c t system of S-modules indexed by a set I of c a r d i n a l i t y < . Then p . d . ^ l i n ^ { A j } T < sup {p.d- 5 A- ) r *af 1* Let {G;} x be a d i r e c t system of groups and G - l i a , { G, j r . Let H be a r i n g . Of course one can take the d i r e c t l i m i t lim^ f A£ j x of a d i r e c t system of BG-modules ( A £ } x to get an BG-module. But there i s another type of d i r e c t l i m i t which we are i n t e r e s t e d i n : Let { Mjjj. be a d i r e c t system of B-modules over the d i r e c t system of groups £ G ^ j j . (This means that Mi i s an BG;.-module f o r each i i n I , t h a t { M-}j i s a d i r e c t system of B-modules, and that the BG;-module s t r u c t u r e s are compatible with the maps i n the d i r e c t systems { G t-j z and £ M tj x .) Then the B-module M = lim., f -H(lj has a n a t u r a l BG-module s t r u c t u r e . To d i s t i n g u i s h t h i s BG-module from the u s u a l d i r e c t l i m i t of BG-modules, we denote i t by Lim > £ Mi3 x. Let £ M j j x and { G-J x be as i n the preceeding paragraph. Then { BG B^.H^jj i s a d i r e c t system of BG-modules, where G = link, {G^.Jj... A r e s u l t of Gruenberg says that l i m £ EG ®^M;J Z = Lim^ ( Mt'3x as BG-modules £ 1, V Lemma 2 j . 12.2. P r o p o s i t i o n . Let { <-i}z be a d i r e c t system, whose maps are i n c l u s i o n s , of subgroups of a group G, such that G = l i % £ G j } r - Suppose the c a r d i n a l i t y of I i s < Then: (1) For any BG-module H, p.d.^M < sup [p. d. M}r • n • • 1. (2) l.gl.d.BG < sup { l . g l . d . BGi } J + n + 1. Proof. (1) Since G i s a subgroup of G, M i s an BG--module f o r each i . Thus we have a d i r e c t system { M }j of B-modules over the d i r e c t system £ G j j j . ,. By Gruenberg's r e s u l t , M = Lirn^ f H J r = lirn^ (RG j r - , By Osofsky*s Theorem, p.d.^M < sup {p.d.^RG ®^,a j r * n • 1. But BG i s a f l a t ( r i g h t ) BG^-module. Therefore p. d.^BG < p.d.^M f o r each i . The proof i s completed. (2) i s an immediate conseguence of (1). 12.3. P r o p o s i t i o n . Let G be a l o c a l l y f i n i t e group of c a r d i n a l i t y < $n . Let B be a r i n g * (1) l.gl.d.BG i s f i n i t e i f and only i f l.g l . d . B i s f i n i t e and G i s B - t o r s i o n - f r e e . (2) I f G i s B - t o r s i on-free, then l.gl.d.B l.gl-d.BG < l . g l . d . B + n • 1. Proo. G i s the d i r e c t l i m i t of the d i r e c t system of 68 f i n i t e l y generated subgroups. The order of every f i n i t e l y generated subgroup i s a u n i t i n B i f and only i f G i s R-t o r s i o n - f r e e . The r e s u l t f o l l o w s from P r o p o s i t i o n s 12.2(2) and 11.1. 12.4. P r o p o s i t i o n . L e t G be a l o c a l l y M-group of c a r d i n a l i t y < £(n , and B be a r i n g . I f every f i n i t e l y generated subgroup of G has a standard M-series, such that the order of i t s f i n i t e f a c t o r i s a u n i t i n R, then: l . g l . d . f i < l.gl.d.BG < l - g l . d . B • h (G) .• n > 1. Proof., Under the assumptions, we have, f o r every f i n i t e l y generated subgroup H of G, l.gl.d.BU < 1 . g l . d.B + h(G). Osofsky^s Theorem i m p l i e s the desired i n e q u a l i t y . Again, the preceeding r e s u l t has a stronger v e r s i o n when G i s l o c a l l y n i l p o t e n t . We f i r s t make e x p l i c i t some pr o p e r t i e s of l o c a l l y n i l p o t e n t groups which w i l l be u s e f u l l a t e r , 12-5. P r o p o s i t i o n . L e t G be a l o c a l l y n i l p o t e n t group, and T be the s e t of i t s t o r s i o n elements* (1) T i s a l o c a l l y f i n i t e l o c a l l y n i l p o t e n t c h a r a c t e r i s t i c subgroup of G., (2) h(G) = h (G/T). (3) I f G i s l o c a l l y n i l p o t e n t of c l a s s l e s s than or equal to a f i x e d number k, then G i s n i l p o t e n t of c l a s s l e s s 69 than or equal to k. Proof. (1) G i s a c t u a l l y l o c a l l y f i n i t e l y generated n i l p o t e n t by d e f i n i t i o n . I t f o l l o w s from P r o p o s i t i o n 11.6 (3) t h a t T i s a l o c a l l y f i n i t e group. I t i s a l s o c l e a r that T i s l o c a l l y n i l p o t e n t and c h a r a c t e r i s t i c . (2) To each f i n i t e l y generated subgroup a of G, there corresponds a f i n i t e l y generated subgroup flT/T of G/T of equal H i r s h number (h (H) = h (H/H D T) = h (HT/TJ) . The converse a l s o holds*. Therefore h(G) = h (G/T) ; by t a k i n g suprema. (3) Induce on k. Suppose k = 1. G i s l o c a l l y a b e l i a n , hence a b e l i a n . Suppose the a s s e r t i o n has been proved f o r groups l o c a l l y n i l p o t e n t of c l a s s < k, and G i s a group l o c a l l y n i l p o t e n t of c l a s s < k. Then by i n d u c t i o n £G,Gj i s a n i l p o t e n t group of c l a s s < k. Therefore G has a f i n i t e lower c e n t r a l s e r i e s of length < k. 12*6. P r o p o s i t i o n . Let G be a l o c a l l y n i l p o t e n t group with c a r d i n a l i t y < . Let T be the t o r s i o n subgroup of G. The f o l l o w i n g a s s e r t i o n s are e q u i v a l e n t : (1) l.gl.d.RG i s f i n i t e . (2) l . g l . d . f i i s f i n i t e , G/T i s n i l p o t e n t , h(G) i s f i n i t e and G i s B - t o r s i o n - f r e e . (3) l . g l . d . f i i s f i n i t e , G/T i s n i l p o t e n t and c.d.^G i s f i n i t e . (4) l . g l . d . f i i s f i n i t e , h(G) i s f i n i t e and G i s a-70 t o r s i o n - f r e e . (5) l . g l . d . B i s f i n i t e , c.d.^B i s f i n i t e . Proof. We f i r s t observe that h(G) < c.d.^G, and t h a t , i f G i s B - t o r s i o n - f ree, c. d.^ G < h (G) • n • 1. To see t h i s , l e t { Gj,jj be the d i r e c t system of the f i n i t e l y generated subgroups of G. Then h(G) = sup £ h(G;) } z < sup { c.d.^Gj J x < c.d.gG. Suppose G i s B - t o r s i o n - f r e e . By P r o p o s i t i o n s 12.2(1) and 11 . 7(2), c.d.ftG < sup { c , & . R G i } z • n + 1 - sup { h(G;) j x • n • 1 = h (G) • n «• 1. Of course, when G i s not B - t o r s i o n - f r e e , c.d.^G i s i n f i n i t e . , I t f o l l o w s from t h i s observation t h a t c.d. RG i s f i n i t e i f and only i f h(G) i s f i n i t e and G i s B - t o r s i o n - f r e e . Therefore (2) i s eq u i v a l e n t t o ( 3 ) , and (4) i s equi v a l e n t t o (5). By P r o p o s i t i o n 12-5(2), h (G) i s f i n i t e i m p l i e s h (G/T) i s f i n i t e * Being t o r s i o n — f r e e , G/T i s l o c a l l y n i l p o t e n t of c l a s s l e s s than o r equal to h (G/T). By P r o p o s i t i o n 12.5 (3), G/T i s n i l p o t e n t provided h(G) i s f i n i t e . I t remains to show that l.gl.d.BG i s f i n i t e i f and only i f l . g l . d . B and c.d.^G are both f i n i t e . Obviously, l.gl.d.BG i s f i n i t e i m p l i e s both l.gl.d.B and c. d.^G are f i n i t e . The reverse i m p l i c a t i o n f o l l o w s from the f o l l o w i n g i n e q u a l i t y . 12.7. P r o p o s i t i o n . Let G be a l o c a l l y n i l p o t e n t group of c a r d i n a l i t y < tfn . Suppose a l s o t h a t G i s B - t o r s i o n - f r e e . , Then h(G) < c.d.^G < l . g l . d. BG < l.gl.d.B * h (G) + n + 1 71 < l . g l . d . B + c.d.^G + n * 1. Proof. He have shown that h<G) < c.d.^G. Let { Gt-.}r be the d i r e c t system of the f i n i t e l y generated subgroups of G. By the r e s u l t s we have proved so f a r , l.gl.d.BG < sup { l.gl.d.BG^ j r • n • 1 < sup { l . g l . d.B+h (G;) }j * n + 1 = l . g l . d . B + h(G) n +1. The proof i s completed. 72 REFERENCES 1 Anderson, F.W. and F u l l e r , K . f i . , Bings and Categories of nodules, Grad. Texts i n Math., v o l 13, Springer-V e r l a g , New York 1973. 2 Arora, S.K. and P a s s i , I.B.S., A n n i h i l a t o r I d e a l s i n Twisted Group Bings, J . London Math. Soc. (2) 15 (1977), 217-220. 3 Balcerzyk, S., The G l o b a l Dimension of the Group Bings of a b e l i a n Groups, Fund. Math. 55 (1964),293-301. 4 Cartan, fi. and E i l e n b e r g , S., Homological algebra, P r i n c e t o n Univ. Press, P r i n c e t o n 1956. 5 Chouinard, L.G., P r o j e c t i v i t y and R e l a t i v e P r o j e c t i v i t y over Group Bings, J . Pure Appl. algebra 7 (1976), 287-302-6 E i l e n b e r g , S., Rosenberg, A- and Z e l i n s k y , D.r On the Dimension of Modules and Algebras V I I I , Nagoya Math. J . 12 (1957), 7fc93. 7 F a i t h , C., Algebra I I , King Theory, Grund. Math. Wissen., v o l 191, S p r i n g e r - V e r l a g , New York 1976-8 F a r r e l l , F-T. and Hsiang W.C., a Formula f o r K( (R^fT]) , Proc. Sympos. Pure Math., v o l 17, amer. Math. S o c , Providence (1970), 192-218. 9 Formanek, E., Maximal Quotient Bings of Group Bings, P a c i f i c J . Math. 53 (1974), 109-116.,, 10 Golan, J.S., L o c a l i z a t i o n of Noneommutative Bings, Marcel Dekker, I n c . , New York 1975. 11 G o l d i e , a.W., Semi-^prime Bings with Maximum C o n d i t i o n s , Proc. London Math. Soc. (3) 10 (1960), 201-220. 12 Goldman, O., Bings and Modules of Quotients, J. algebra 13 (1969), 10-47. 13 Goodearl, K.B., S i n g u l a r Torsion and the S p l i t t i n g P r o p e r t i e s , Memoirs amer.. Math. Soc. , No. 124 (1972) . 14 Gruenberg, K.H., Cohomological Topics i n Group Theory, Lecture Notes i n Math., v o l 143, S p r i n g e r - V e r l a g , New York 1970. 73 15 H a l l , P., N i l p o t e n t Groups, Queen Mary College Math-Notes 1969-16 H e r s t e i n , I.N. and Small, L.W., Rings of Quotients of Group algebras, J . Algebra 19 {1971), 153-155., 17 H i l t o n , P.J. and Stammback, 0., A Course i n Homological alg e b r a , Grad. Texts i n Math., v o l 4, S p r i n g e r - V e r l a g , New York 1971. 18 Jans, J.P., Bings and Homology, H o l t , fiinehart and Winston, New York 1964. 19 Lambek, J . , Lectures on Bings and Modules, B l a i s d e l l Pub. Comp., Waltham 1966.. 20 Lambek, J . , Torsion Theories, a d d i t i v e Semantics and Bings of Quotients, Lecture Notes i n Math., v o l 177, S p r i n g e r - V e r l a g , New York 1971. 21 Lambek, J . , Bicommutators and Nice I n f e c t i v e s , J . algebra 21 (1972), 60-73. 22 Lambek, J . and M i c h l e r , G., The Torsion Theory a t a Prime I d e a l of a Bight Noetherian Ring, J . Algebra 25 (1973), 364-389. 23 Lambek, J . and M i c h l e r , G., L o c a l i z a t i o n of B i g h t Noetherian Bings at Semi-prime I d e a l s , Canad. J . Math. 26 (1974), 1069-1085. 24 Louden, K., Torsion Theories and Bing Extensions, Comm. algebra 4 (1976), 503-532. 25 Louden, K., Maximal Quotient Bings of Bing Extensions, P a c i f i c J . Math. 62 (1976), 489-496. 26 MacLane, S., Categories f o r the Working Mathematician, Grad. Texts i n Math., v o l 5, Springer-rVerlag, New York 1971. 27 Murdoch, D.C.,and Oystaeyen P., Symmetric Kernel Functors and Quasi-Primes, Indag. Math. 37 (1975), 97-104. 28 Murdoch, O.C. and Oystaeyen F., Non-Commutative L o c a l i z a t i o n and Sheaves, J . algebra 35 (1975), 500-515. 29 Osofsky, B.L., Homological Dimension of Modules, Regional Conference S e r i e s i n Math-, v o l 12, amer. Math. S o c , Providence 1973. 30 Passman, D.S., The a l g e b r a i c S t r u c t u r e of Group Rings, 74 John R i l e y and Sons, New York 1977. 31 Bim, D.S., Modules over F i n i t e Groups, Ann. Of Math. 69 (1959), 700-712. 32 fiotoan, J . J . , The Theory of Groups, second e d i t i o n , A l l y n and Bacon, I n c . , Boston 1973. 33 S c o t t , W.B., Group Theory, P r e n t i c e - H a l l , Inc., Hew Jersey, 1964. 34 Stenstrom, B., Bings and Modules of Quotients, Lecture Notes i n Math., v o l 237, s p r i n g e r - V e r l a g , New York 1971. 35 W a r f i e l d , B.B.Jr., N i l p o t e n t Groups, Lecture Notes i n Math., v o l 513, S p r i n g e r - V e r l a g , New York 1976. 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            data-media="{[{embed.selectedMedia}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
https://iiif.library.ubc.ca/presentation/dsp.831.1-0080276/manifest

Comment

Related Items