UBC Theses and Dissertations

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UBC Theses and Dissertations

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

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TOPICS 08 NQNCOHMUTATIVB LOCALIZATION AND GROUP RINGS  by  KIT-SUM |LEE B.Sc, M.A.,  The C h i n e s e U n i v e r s i t y of Hong Kong, 1968 U n i v e r s i t y o f New Brunswick, 1973  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENT FOR THE DEGREE OF  D0CT08 Of PHILOSOPHY in THE  FACULTY  OF  GRADUATE  STUDIES  ( i n t h e 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 )  In  presenting  an  advanced  the I  Library  further  for  his  of  this  written  shall  agree  The 2075  thesis  in  at  University  the  make  that  it  thesis  purposes  for  partial  freely  permission may  representatives.  be  It  of  University Wesbrook  for  gain  Vancouver,  of  British  Canada  1W5  March 1 6 , , 1 9 7 9  Columbia  for  extensive by  shall  of  the  requirements  B r i t i s h Columbia,  the  understood  Mathematics  Place  of  granted  is  financial  fulfilment  available  permission.  Department  V6T  degree  scholarly  by  this  copying Head  that  not  reference  of  copying  be a l l o w e d  agree  and  of my  I  this  that  study. thesis  Department or  "for  or  publication  without  my  ABSTRACT  S u p e r v i s o r : S. s.  Page  Three t o p i c s i n  localization  theory  and  group  ring  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  ring  is  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 r e s p e c t t o a prime k e r n e l f u n c t o r t o be s e m i - s i m p l e is  found.  &n  analogous  functors i s obtained  result  i n Section  In Chapter I I , t h e i d e a applied  to  group  ring  for  of  Section  the  classical  kernel  5.  of  controlling  localization.  ring  Artinian  guasi-prime  subgroups  Some  of  is  sufficient  c o n d i t i o n s f o r t h e descent o f the maximal r i n g of and  with  quotients  quotients are obtained  in  8.  I n C h a p t e r 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 left  global  application,  dimension this  is  finite  homological  i n f o r m a t i o n on the t o r s i o n  are  result  obtained. is  used  e l e m e n t s o f an M-group.  As to  an get  iii TABLE OF CONTENTS  CHAPTER I . NONCOMMOTATI7E LOCALIZATION  -  .1  S e c t i o n 1. T o r s i o n Theory ..........................  1.  S e c t i o n 2. R i n g s and Modules o f Q u o t i e n t s ..........  7  S e c t i o n 3. Examples ................................12 S e c t i o n 4. Prime K e r n e l F u n c t o r  17  S e c t i o n 5. Q u a s i - p r i m e K e r n e l F u n c t o r s ............. 25 CHAPTER I I . LOCALIZATION OF GROUP RINGS .....  34  S e c t i o n 6. The C o n t r o l l e r .......................... 34 S e c t i o n 7. The R i n g s Of Q u o t i e n t s Of A Group B i n g . . 4 0 Section  8.  The  Maximal  S i n g Of Q u o t i e n t s And The  C l a s s i c a l K i n g Of Q u o t i e n t s  48  CHAPTER I I I . GLOBAL DIMENSION OF GfiOUP RINGS  53  S e c t i o n 9. Background .............................. 53 S e c t i o n 10. The Commutative Case ................... 59 S e c t i o n 11. The F i n i t e  Case  And  The  Case  Of  M-  gr oups ...... - - ...... ............... ... . -- .... . - »•, 61 Section  12. O s o f s k y * 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.  . . . . , . . . . . . . .  72  ACKNOWLEDGEMENTS  The  author  supervisor, tendered  wishes  Dr- S.S-  t o express Page,  h i s thanks  to h i s  f o r encouragement and a d v i c e  d u r i n g the e x e c u t i o n o f p a r t o f the r e s e a r c h  herein  described. The a u t h o r wishes t o thank Dr. D.C- Murdoch, h i s f o r m e r s u p e r v i s o r , f o r a s s i s t a n c e d u r i n g t h e e x e c u t i o n of  most  of  the research described i n Chapter I . The  f i n a n c i a l support o f t h e 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  SectionJ.  Torsion  Theory  In t h i s s e c t i o n torsion  theory.  we s h a l l o u t l i n e  &  the basic  facts  t o r s i o n theory i s characterized  of  i n four  ways: (1). by a c l a s s o f t o r s i o n modules; (2) by a  equivalent  c l a s s o f t o r s i o n - f r e e modules; (.3) by an (H)  functor;  by  a  idempotent  kernel  topology of l e f t i d e a l s . Every t o r s i o n  t h e o r y i s generated by a module and c o g e a e r a t e d by a module. The s e t o f a l l t o r s i o n t h e o r i e s ring)  (with  respect  to a  fixed  i s equipped w i t h a p a r t i a l o r d e r which t u r n s i t i n t o a  complete  lattice.  Let  B  be an a s s o c i a t i v e r i n g w i t h u n i t y . The c a t e g o r y  of l e f t u n i t a r y  fi-modules  no danger o f c o n f u s i o n .  i s denoted by ^ H , o r B i f t h e r e i s  F o r M, N i n M,  the abelian  group  Hom (M,N)  i s denoted by £M,N1^, o r £M,NJ. I(M) denotes t h e  injective  hull  /(?  monomorphism, means f  of  H  i s aa  F  To  say f  i n [M,NJ i s a  epimorphism..,. L e t T  and  F  be  H and H be i n M. By £a,F]=0, we mean £H,F.]=0  every F i n F. By  every  i n H.  we s i m p l y w r i t e f:M>—>N. S i m i l a r l y , f:H—>>N  i n £M,Nj  subclasses for  of  i n F.  £M,I(F) j=0,  £T H]=0 #  we  mean  £M,I{F)]=0 f o r  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 every  C o f M i s s a i d t o be c l o s e d under e x t e n s i o n  i f for  s h o r t e x a c t sequence M > — > N — » L i n M, M and L a r e i n  C i m p l i e s N i s i n C. Throughout t h i s paper, t h e word " i d e a l " w i l l  mean  two  s i d e d i d e a l , u n l e s s s p e c i f i e d by " l e f t " o r " r i g h t " ; the word "module" r e f e r s t o l e f t  1.1.  Proposition.  subclasses  of  £20,  M,  module.  Proposition  closed  under  0.3}  Let T  isomorphic  and F be  images.  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)  T  i s closed  under q u o t i e n t s , e x t e n s i o n s ,  direct  sums and submodules; and .£= {M |[T, M j=0] . (2) F i s c l o s e d under  submodules,  extensions,  direct  p r o d u c t s 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=  1. 2. _Def i n i t i o n . (T,F)  torsion_theory,  ]=0j.  i s a p a i r of  subclasses  s a t i s f y i n g t h e 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 as  A  £ H | [ T > I ( H )  (T,F) be a t o r s i o n t h e o r y , and M be i n M. D e f i n e  K  t h e sum of a l l submodules o f M which b e l o n g t o T, and L  as t h e i n t e r s e c t i o n o f a l l submodules N o f M such  that  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  Definition.  An  idempptent k e r n e l f u n c t o r  as IKF) i s a s u b f u n c t o r that:  (abbreviated  t of t h e i d e n t i t y f u n c t o r of M  such  (1) t(M/t(M))=0 f o r e v e r y H i n M, and (2) i f M<N i n H,  3 then t (M) =M f) t (N). a  submodule  I n o t h e r words, t a s s i g n s t o e v e r y M i n M  t(M)  and  to  every  M,  N  £ ia, N J — > [ t <M) , t (H) J by r e s t r i c t i o n , such t h a t are  M  (1)  a  map  and  (2)  satisfied.  1.4. P r o p o s i t i o n . correspondence  Let is  in  {34,  Corollary  2.7] There i s an one-one  between t o r s i o n t h e o r i e s and I K F * s .  (T,F) be a t o r s i o n t h e o r y , t h e c o r r e s p o n d i n g IKF  t  d e f i n e d by : t(M) = t h e 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 o f a l l  the  submodules  which b e l o n g t o T. C o n v e r s e l y , t o an IKF t t h e r e c o r r e s p o n d s a  torsion  theory  (T,F) such t h a t M i s i n T i f and o n l y i f  t(M)=H, and M i s i n F i f and o n l y i f t ( f l ) = 0 .  1.5. D e f i n i t i o n . A t o p o l o g y i s a non-empty f a m i l y D of  left  i d e a l s o f 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, r x 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  is  i n D f o r every a i n A, then B i s i n D.  It  can  readily  be  s a t i s f i e s the f o l l o w i n g  proved  that  a  topology  D also  properties:  (3) I f A i s i n D and B i s a l e f t  ideal  containing  A,  then B i s i n D. (4)  If  A,  B  are i n D, then t h e i n t e r s e c t i o n and the  4 p r o d u c t o f A and B a r e i n D. When (1) , (3) and (4) h o l d , B with  i s a  topological  ring  D as a fundamental system o f neighborhoods o f t h e z e r o  element.  1.6..^proposition.  £34, Theorem  3.4]  There  i s an  one-one  c o r r e s p o n d e n c e between I K F ' s and t o p o l o g i e s .  The  correspondence  assigns  t o an IKF t t h e t o p o l o g y  D = {D J t(B/D)= B/Dj, and t o a t o p o l o g y D t h e I K F t such t(M)  equals  that  t h e s e t of e l e m e n t s 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 t h e r e i s a one-one  correspondence  t h e o r i e s , I K F * s and t o p o l o g i e s . be  the corresponding  among  torsion  L e t t be an I K F , (T,F) and D  t o r s i o n t h e o r y and t h e c o r r e s p o n d i n g  t o p o l o g y , 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 fi 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 o f 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 t h e f a m i l y  of  a l l t-dense  left ideals.  1.7. P r o p o s i t i o n .  £10,  Proposition  IKF^s t ( i ) c o r r e s p o n d t o the and  the topologies  torsion  D ( i ) . The  i s  c o n t a i n e d i n Tj[2}_.  (2) l l l l . c o n t a i n s  F 12) .  theories  following  equivalent; (1) - -HIL  7.1 J F o r 1=1,2, l e t t h e ( T ( i ) ,F ( i | >  conditions  are  5  (3)  F o r each M i n M, t ( 1 ) <M) i s c o n t a i n e d i n t{2) (13). Mil  i  s  c o n t a i n e d i n U.12.L-  \  1.8. . D e f i n i t i o n . The IKF t ( 1 ) i s s m a l l e r than t ( 2 ) , o r t ( 2 ) is  g r e a t e r than  tlll.£%12L  t ( 1 ) , or  *  i f  only i f the  a n d  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 a r e s a t i s f i e d .  1.9. D e f i n i t i o n . L e t C be a torsion  theory  which  of  M.  The  smallest  f o r which C i s c o n t a i n e d i n T i s c a l l e d t h e  t o r s i o n t h e o r y generated for  subclass  by C. The g r e a t e s t  torsion  theory  C i s c o n t a i n e d 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 , are  they  way £20, S e c t i o n O j ;  For a s u b c l a s s X of M, l e t 1 (X) = {H|1X,I(H) 3=0j.  r l (X) c o n t a i n s X, and X c o n t a i n s r (1) Let  Then  = [M|£ M,I (X) ]=0j  obviously  i s contained  in Y  implies  C  be  a  subclass  of  H. Then r l r (C) =r ( l r (C)) i s r (C) .  (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 generated  S i m i l a r l y , ( 1 ( C ) . , r l (C) ) i s t h e unique t o r s i o n  cogenerated  Thus  Similarly, lrl(C)=l(C).  t h e o r y . I t i s o b v i o u s l y t h e unique t o r s i o n t h e o r y C.  r (X)  and 1{X) c o n t a i n s 1 (Y) .  rlr(C)=r(C).  by  and  l r { X ) c o n t a i n s X,  c o n t a i n e d i n r(C) , and r l r (C) = r l (r (C)) c o n t a i n s  By  exist,  unique? Both q u e s t i o n s a r e 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  r(X)  i f they  by  C.  Moreover,  l r(C)  = {MJ£fl,r(C) ]=0] ,  theory and  6 rl(C)  = (H|£l<£) ,M]=0}. a  consequence  of  this  observation  i s the f o l l o w i n g  Proposition.  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  t o r s i o n t h e o r i e s i s a complete  8.1}  The  set  lattice.  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 t h e o r y . O)  of a l l  Then  C2*£) i s generated by a s i n g l e module, namely, t h e  d i r e c t sum o f a l l R/&,  where  A  varies  over  dense  left  ideals. (2)  is  cogenerated by a s i n g l e module, namely,  t h e d i r e c t p r o d u c t o f a l l R/A,  where k  varies  over  closed  left ideals. (3) R/A,  (1#D  i s a l s o c o g e n e r a t e d by t h e d i r e c t sum of a l l  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. generated  (1) L e t t=(T,£J and by  the  direct  sum  m  be  the  torsion  theory  o f H/A*s, where A i s dense.  C l e a r l y t>a. Suppose t#u, then t h e r e i s a  left  ideal  which i s t-dense but not m-dease. L e t ffl<B/A)-B/A, where  A  ,  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 t h e 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  S e c t i o n 2. flings and Modules o f Q u o t i e n t s  In  this  s e c t i o n we s h a l l d e s c r i b e t h e c o n s t r u c t i o n o f  q u o t i e n t s , which i s t h e b e s t commutative torsion  localization  theory i s  localizations,  a  but  known  generalization  technique.  property  shared  of  the  The  perfectness of a  by  a l l commutative  not n e c e s s a r i l y r e t a i n e d i n the g e n e r a l  case.  Let and  (T,F) be a t o r s i o n  corresponding  t h e o r y w i t h c o r r e s p o n d i n g IKF  t o p o l o g y D. D i s a d i r e c t e d s e t w i t h i t s  p a r t i a l o r d e r d e f i n e d by i n c l u s i o n . Q {F) t  = Q{F)  t  = li^{£0#FJlD  F o r each F i n F,  define  i s i n D} a s the d i r e c t l i m i t o f  t h e 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 o f Q(F) i n t h e f o l l o w i n g way: L e t r be i n fi and x i n is  Q(F).  r e p r e s e n t e d by some f i n J.D,F}. D e f i n e r x as t h e element  i n Q(F) r e p r e s e n t e d by g i n £ £ D:r J,F ], where g (s)=f (sr) . is  x  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  and t h a t Q(F) becomes an H-module w i t h t h i s For  M  in  H,  define  It  i s well defined, multiplication.  Q (M) = Q<M/t(M)).  It  is  not  d i f f i c u l t t o see t h a t 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)  is  c a l l e d t h e module of q u o t i e n t s ,  w i t h r e s p e c t t o the IKF t , o f M.  2.2. D e f i n i t i o n , for  any  a module E i n M i s c a l l e d  t-injective i f ,  M, N i n H such t h a t u i s c o n t a i n e d i n M and M/N i s  8 t - t o r s i o n , t h e 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 t h e 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 a r e 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) F o r 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 a r e 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-injective.  (,2) E i s t - i n j e c t i v e and t - t o r s i o n - f r e e . (3) E i s t - t o r s i o n - f r e e , and, f o r each  t-torsion-  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 . isomorphism,  the  £12, unique  Theorem  3.7J Q(M) ; i s ,  faithfully  t-injective  up  to  module  satisfying: (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) a s s i g n s m i n H/MM)  t o f - i n .£B,-fl/t(H>.J  such t h a t f ( r ) = r m .  2.5. P r o p o s i t i o n . ( 2 0 , P r o p o s i t i o n torsion-free  module  = p-i ( t ( I / T H , ;  where  with  0.7]  injective  p: I — > I / F  is  Let F  be  hull  I.  Then  the  natural  a tQ (F) 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 a r e now ready t o d e f i n e a r i n g s t r u c t u r e on Q(B). L e t x, y be elements of Q (H) . , D e f i n e f i n £B,Q(B) ] by f ( r ) = r y . S i n c e 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 t o g i n £Q(S) »Q(H) J u n i q u e l y .  L e t 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 c o m p a t i b l e with i t s B-module s t r u c t u r e . It  i s clear  that  t (B)  i s an i d e a l . I t f o l l o w s  that  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 (with respect  t o t) .  2.6.  Proposition.  left  exact.  In IKF t  the r i n g of quotients  J. 12, Theorem 3.9] The f u n c t o r  i s called  Noetherian  f o l l o w i n g c o n d i t i o n s : Bhenever A <  1  2  Q: H—>M i s  i f i t s a t i f i e s the  > < . » . i s an a s c e n d i n g  c h a i n o f l e f t i d e a l s whose u n i o n i s t - d e n s e , t h e r e i s some n s u c h t h a t a<n> i s t-dense. a H—»N  module  P  i s t-projective  o f t o r s i o n - f r e e modules  i f f o r any epimorphism  and any map  modules, t h e r e i s a dense submodule P  f  P—>N  o f B-  o f P and a map P'—>M  of modules s u c h t h a t the f o l l o w i n g diagram i s commutative.  10  2.7. P r o p o s i t i o n .  pt>  >p  H  »N  £ 12,  Theorem 4.4 J An IKF t i s Hoethe.ri.an  i f and o n l y i f t h e f u n c t o r Q: M—>H p r e s e r v e s 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 i f and o n l y i f e v e r y t-dense l e f t  Let  A  denote  the f u l l  ideal i s ^projective.  subcategory  in M  of  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 It  CQ in  » 3 ^ ^ £M,A3 a  R  A.  However,  in H  and e v e r y  and t h e isomorphism  the inclusion  functor  A  i n A,  i s n a t u r a l i n M and  f u n c t o r A—>M does n o t , i n  g e n e r a l , p r e s e r v e c o p r o d u c t s and epimorphisms. .For if  example,  A and B a r e i n A such t h a t B i s a submodule o f A and A/B  i s i n T, then t h e i n c l u s i o n B—>A i s an epimorphism but of  i n A,  i s n o t n e c e s s a r i l y e p i c i n M. Thus t h e l e f t a d j o i n t a e s s Q: M—>A i s n o t t o o i n f o r m a t i v e . Be t u r n o u r a t t e n t i o n  modules.  embedding  t o t h e category  0  of  Q(R)—  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,  Corollary  4-23-  Thus  there  i s a  A — > £ . The r i n g map B—>Q(B) i n d u c e s a n o t h e r  embedding Q.—>M. C l e a r l y , the c o m p o s i t i o n f u n c t o r is  a l l  functor-  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  A—>H* T h i s means, f o r every M  exact  the natural  full full  A—>Q—->I  embedding o f A i n t o M. I t i s i n t h i s sense  11  t h a t we c o n s i d e r contained  2.9.  A and Cj as f u l l s u b c a t e g o r i e s  o f M, w i t h  A  i n 0,.  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 a r e e q u i v a l e n t ; (1) Q i s c o n t a i n e d (2)  F o r every  i n F. i d e a l D, Q (S) i (D) = Q (fi) ,  t-dease l e f t  where i ; B — > Q (fi) i s the n a t u r a l map. =  (3)  (4)  A.  Q(B)8 _: M—->£  equivalent  and  Q: M—>0.  are  naturally  functors.  <5) Q: H—>M i s r i g h t e x a c t and p r e s e r v e s d i r e c t sums. (6)  Q: M—>p. i s t h e l e f t a d j o i n t o f t h e f u l l  2 — > I , w i t h t h e ad j u g a n t  [ Q (M) ,D  — > £ H,.D J  R  embedding  defined  by  restriction.  Proof.  The e q u i v a l e n c e o f (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 It  i s well  known  that  Q(fi) ®^_: M—>Cj. T h e r e f o r e  2.,10.,, D e f i n i t i o n .  An  J. B,_ j : Q—>M  t o i R,_ J.  i s right adjoint to  (4) and (6) a r e e q u i v a l e n t .  IKE t  i s called  perfect  assertions i n Proposition 2.9 are s a t i s f i e d . ,  i f the  12  S e c t i o n 3.  Examples  F i v e examples a r e i n c l u d e d i n t h e 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  discussion.  3.1.  D e f i n i t i o n . A t o r s i o n theory  i s called faithful i f 1 i s  torsion-free. ,  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 contains  be a m u l t i p l i c a t i v e l y c l o s e d s u b s e t of a,  C  the u n i t y 1 but not 0.  Then  the  family  which  of  left  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  condition  is  s a t i s f i e d : For e v e r y c i n C, and e v e r y r i n fi, t h e r e are  c'  in  C  and  if  and  only  if  the  following  r* i n E, such t h a t 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 t h e IKF d e f i n e d above i s  denoted  by /1(C). Let  C(fi)  be  the  sided non-zero-divisors Ore.  multiplicatively of H.  c l o s e d s e t of  Suppose a l s o t h a t C (R) i s l e f t  Denote t h e IKF /t(C (R)) by c l . Then (1) F o r every H i n H, c l (Mj i s the s e t of a l l  m of M which are a n n i h i l a t e d by some c i n (2) c l i s f a i t h f u l and (3)  Q(S)  that  elements  C(R).  perfect.  i s t h e 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  That i s t o say: t h e r e i s an embedding o f r i n g s such  two  i (c)  i s a u n i t i n Q(R)  R.  i : R—>Q(R),  f o r e v e r y c i n C(R) ,  and  13 t h a t e v e r y q i n Q(B) i s e q u a l t o i ( c ) C(8)  - l  i(r)  f o r some  c in  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 o f H  i s Z(M) = £m|m i s i n a , t h e a n n i h i l a t o r o f m i s e s s e n t i a l i n Bj.  a i s called singular  i f Z(a)=8,  and  non-singular i f  z (a) =0.  3.4. Example- The G o l d i e t o r s i o n t h e o r y . The  class  of P r o p o s i t i o n  F o f a l l aon-^singular modules s a t i s f i e s (2) 1-1. T h e r e f o r e F i s t h e c l a s s o f t o r s i o n - f r e e  modules f o r some t o r s i o n called IKF  the  theory.  This  torsion  theory  i s  G o l d i e t o r s i o n t h e o r y , and the c o r r e s p o n d e n d i n g  i s denoted  by  g o l . Some  observations  and  other  characterizations are; (1)  Let H  be i n B . .2(H) i s an e s s e n t i a l submodule o f  g o l (H) £34, P r o p o s i t i o n 3 . 5 ] . (2) torsion  S i n c e Z (fl) i s s i n g u l a r f o r each U i n B, the theory  i s generated  by t h e c l a s s o f 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 topology containing (3) g o l ( a ) / Z ( a ) (4)  Goldie  i s the  smallest  the family of a l l e s s e n t i a l l e f t i d e a l s . = Z(B/Z(a))  £34, P r o p o s i t i o n 3.5J.  I t f o l l o w s from (3) t h a t a module a i s g o l - t o r s i o n  i f and o n l y i f H/Z(a) i s s i n g u l a r . I n p a r t i c u l a r , I(H)./H gol-torsion  f o r every  h u l l of a . By P r o p o s i t i o n a i n a.  is  a i n a , where 1 ( a ) i s t h e i n j e c t i v e 2.5, Q(a) = I (S/gol (M) ) f o r e v e r y  14 _BaazZg£9~diy4^Q3?s o f B i s ^ a  (5) Suppose t h e set,, g i g ! l e f t Ore s e t . Then i n C (B), the  the IKF c l i s d e f i n e d , and, f o r e v e r y c  left  ideal  Rc  is  gol-dense.  Consequently,  c l < gol* Suppose contains left  further  that  a non-zero-divisor.  Goldie  every e s s e n t i a l l e f t i d e a l  Then c l = g o l . I n a  semi-prime  r i n g , every e s s e n t i a l l e f t i d e a l does c o n t a i n  a  non-zero-divisor.  3.5. Example. The maximal l e f t  ring_of-_guotieat.  The t o r s i o n t h e o r y c o g e n e r a t e d by fi, o r ,  equivalently,  by 1 ( B ) , i s c a l l e d t h e Lambek t o r s i o n t h e o r y . By d e f i n i t i o n , it  i s the  l a t t i c e of  join  of  IKF's.  a l l f a i t h f u l t o r s i o n theories i n the  The  IKF  corresponding  to  the  Lambek  t o r s i o n t h e o r y i s denoted by max. The r i n g of q u o t i e n t s i s c a l l e d t h e maximal l e f t r i n g o f q u o t i e n t s (1) max < g o l £34, P r o p o s i t i o n (2) free,  Suppose  or,  max = q o l .  It  - Q q0i (B)  f o l l o w s from  = 1(B).  assumption,  gol  D ^,  ax  is  o f B.  3.9J.  B i s non-singular.  equivalently,  Then B i s g o l - t o r s i o n faithful.  Consequently,  (4) o f Example 3.4 t h a t  It  i s also  = D  - the  clear  that,  family  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  Q(B)  of  under  Q^xW this  essential left  Noetherian  semi-prime,  then max = g o l = c l . Let  t be an IKF whose t o r s i o n t h e o r y i s cogenerated by  an i n f e c t i v e module C. L e t H = C # 3^ c  ring  of  C.  C  can  be  c  *>  e  the  endomorphism  c o n s i d e r e d as an B-H bimodule. The  15 endomorphism r i n g B = [ C C j ^ C.  Being  faithfully  module s t r u c t u r e  t-injective,  such t h a t  l c* c~  k (g) (cj.  k: Q£ ( B ) — > B  by  Then k i s an  embedding  = i(t(fij) = 0,  i s c a l l e d the  Qt(f{)~  = gc,  and  has  a natural  ker(kjni(B)  j are  of  0_ (B) ^ t  H. D e f i n e a r i n g  f o r g i n Q (Bj and  because  where i  C  bicommutator  map  c in  C.  = i(Ann (C)) R  the maps i n d i c a t e d  in  the  t  is  commutative diagram below:  fi  j  \  V  >B  /  /  Q (B) t  Hhen C i s a faithful,  finitely it  is  known  P r o p o s i t i o n B j [34, following  M a x  {B)  B, the  torsion  Let  prime  P be  a  cogenerated by the a t P. The  and  [21, the  the  other c  B/P}  =  {cjfor  of  as  rings.  t h e o r y at,;a prime i d e a l . ideal B/P  of  fi.  The  torsion  i s c a l l e d the t o r s i o n  c o r r e s p o n d i n g IKF i s denoted by  i m p l i e s x i s i n P}. family  when  In p a r t i c u l a r , we have  bicommutator of 1 ( H ) ,  fi-module  hand, l e t p:  C(P)••••= C l c i s i n B,  ring  or  k i s a r i n g isomorphism  Theorem 8-4j.  Example. The  On  that  H-module,  result:  (3j Q  3.6.  generated  Tp.  B—>B/P be  theory theory  f  the  natural  p (c) i s a n o n - z e r o - d i v i s o r  of  map the  e v e r y x i n B, cx i s i n P o r xc i s i n P Then i t  is  easy  to  verify  l e f t i d e a l s D = (D|for e v e r y r i n J ,  that  the  £D:rjnc(P)  16 i s non-empty} o f B i s a topology-  The IKF c o r r e s p o n d i n g t o D  i s denoted by /Up . when C (P) i s a =  left  Ore  s e t of  B,  0  { D l D f l C ( P ) i s n o t empty}. (1)  fii <  X.  p  Proof.  p  D i s n o t tp-dense.  Suppose  There i s a non-zero  map f i n £B/D,I (B/P) }. I t f o l l o w s t h a t t h e r e i s an such  0 # f(fx3)  that  x  in  B  i nfi/P.Then £D:x]|f (£x3) = 0 i n fi/P.  T h e r e f o r e £D:x]flC(P) i s empty, and D i s not /^-dense.  (2)  If  every e s s e n t i a l l e f t i d e a l o f t h e r i n g B/P  contains a non-zero-divisor  ff  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 ],  this  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 N o e t h e r i a n r i n g . (3)  Suppose  C(P)  is a  left  Ore s e t of B. Then one  e a s i l y s e e s from t h e c o n s t r u c t i o n o f t h e r i n g s o f that by  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  (2) o f P r o p o s i t i o n 2.9.  3.7. dense  D e f i n i t i o n . A t o r s i o n t h e o r y i s symmetric i f e v e r y  left  ideal  corresponding  to a  contains symmetric  a  dense torsion  symmetric k e r n e l f u n c t o r , a b b r e v i a t e d IKF-  quotients  ideal.  An  IKF  theory i s c a l l e d a  as SKF. L e t t  be  an  The g r e a t e s t SKF l e s s t h a n o r e q u a l 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 t h e o r y a t a prime  ideal. Let defined  P  be  a  prime  i n Example  containing topology.  some  ideal  3 . 6 . The  ideal  of  8, and C(P) be t h e s e t  family  D  of  left  ideals  (c) •= BcB, where c i s i n C<P), i s a  The  corresponding  Since  the i d e a l  I K F , denoted  by  Cp  ,  is  symmetric. <1) C (P) ,  (c) i s ^ - d e n s e f o r e v e r y c i n  Op < jUp < Tp .  (2) Suppose  every e s s e n t i a l  contains a non-zero-divisor.  l e f t i d e a l o f t h e r i n g B/P  Then  0~p  =  f>p°  [27,  =  Example b e f o r e P r o p o s i t i o n 12 j .  S e c t i o n 4. Prime K e r n e l  I n t h i s s e c t i o n the (PKF) due  notion  P  of  prime  kernel  functor  As an immediate consequence o f a r e s u l t  t o Lambek and M i c h l e r , we  ideal Tp  i s defined.  Functor  of a l e f t N o e t h e r i a n  as d e f i n e d i n Example 3 * 6 .  show  that,  to  every  prime  r i n g , t h e r e c o r r e s p o n d s a PKF He a l s o d e s c r i b e a  condition  under which t h e r i n g o f g u o t i e n t s o f B w i t h r e s p e c t t o a PKF i s simple  4.1.  Artinian.  Definition.  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 C i sc r i t i c a l i f  i t i st-critical  with  respect  ideals. to  some  18 IKF t .  In  the  cogenerated left  following,  the  IKF  whose  torsion  theory i s  by a module S w i l l be denoted by *V . i f A i s S  t't/j  ideal,  i s also  a  denoted by Tfi t i f t h e r e i s no  danger o f c o n f u s i o n . Every c r i t i c a l is a critical other  left  words,  l e f t i d e a l C must be t - c r i t i c a l . c  C  i d e a l , then C must be i r r e d u c i b l e , o r , i n  R/C i s a uniform module. I n g e n e r a l , a module  i s uniform i f every module  If  non-zero  submodule  i s essential.  H i s indecomposable i f H has o n l y two t r i v i a l  summands, namely, 0 and M. A module i s uniform i f and  A  direct only  i f i t s i n j e c t i v e h u l l i s indecomposable., A  left  i d e a l A i s prime i f , f o r any x and y i n R, xRy  i s c o n t a i n e d i n A i m p l i e s x i s i n A o r y i s i n A. I f A i s prime  left  ideal,  then  [A:RJ,  the  unique b i g g e s t  a  ideal  c o n t a i n e d i n A, i s a prime i d e a l .  4.3. Definitions  A module S i s a s u p p o r t i n g module o f an IKF  t i f : (1) S i s t-torsion-free, S*#0  o f S, S/S«  and  ( 2 ) f o r any  submodule  i s t-torsion.  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 ,  a b b r e v i a t e d a s PKF, i f t h e r e i s a s u p p o r t i n g module S  of  t  such t h a t t = ^ 5 . It  follows  from t h e d e f i n i t i o n t h a t t i s a PKF i f and  o n l y 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  ideal  19  A.  Let  t  be  isomorphism)  a  PKF,  then  there  is a  t - i n j e c t i v e supporting  Theorem 6 . 4 j . F o r  any  supporting  unique  module module  Q  S  (up  of  of  to  t [12,  t,  Q  is  i s o m o r p h i c t o Qfc(S). . The  following  Proposition  indicates  the  similarity  between PKF and prime i d e a l .  4.5. P r o p o s i t i o n , f 1 Q  #  Proposition  t (1) , t ( 2 ) be I K F * s . Then t = t h e  19.11] L e t t be a PKF and meet  of  t (1)  and  t (2)  i m p l i e s t = t ( 1 ) o r t = t (2).  4.6. P r o p o s i t i o n .  £22,  Theorem  2.13]  Suppose  8  i s left  N o e t h e r i a n . An IKF t i s a PKF i f and o n l y i f t = T , the I K F FI  c o g e n e r a t e d 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 . L e t A be a c r i t i c a l prime l e f t P = £ A: 8 J . Then  Proof. Suppose Since  J = P and  and  T < Tp . n  I t suffices  to  show B/P i s  (B/P) = J/P# where J i s an  £"J/P,R/A3  ideal  = 0,  ideal  Tfi-torsion-free. containing  J i s c o n t a i n e d i n A. I t f o l l o w s  P. that  % (B/P) = 0. ,  4.8. P r o p o s i t i o n .  £22,  Theorem  3.9]  N o e t h e r i a n and P i s a prime i d e a l . Then  Suppose  B  is  left  7^, i s a PKF.  P r o o f . Due t o a r e s u l t o f 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 t h a t £A*:B] = P# and the image o f R/P i n © f c e / B / A ^ i 3.9 ]) .  Therefore  I (R/P)  implies that  tp <  Proposition  4.7,  e v e r y i and  s  =  e s s e n t i a l (see £22, Theorem I(B/A^).  f o r e v e r y i . , On ?/).^  f o r every  This  isomorphism  the other i ^ Thus  hand,  by  T = tfi- f o r p  i s a PKF.  Suppose B i s l e f t N o e t h e r i a n and t A i sa critical independent  prime  left  ideal.  Then  be a PKF, where Q  = Qfc(B/A) i s  o f t h e c h o i c e o f A, and i t can be proved t h a t 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 nonz e r o submodule o f Qj t = Tpi  £22,  Lemma  2.10].. C o n s e q u e n t l y , i f  f o r some prime i d e a l P», then P' must be £A:BJ. I n  the f o l l o w i n g P r o p o s i t i o n , we d e s c r i b e  those PKF  which  are  e q u a l 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 N o e t h e r i a n and A i s a  prime c r i t i c a l l e f t i d e a l . L e t P = £ A:B ]. (1) ^  =  i f and o n l y i f ( A ; x ] / l C { P ) i s empty f o r any  x not i n A. (2) HA =  Proof. the  left  i m p l i e s A f t C ( P ) i s empty.  Since non-zero-divisors Noetherian  prime  form a l e f t Ore  set i n  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}. fA:x]f)C(P)  i s empty  Thus  the condition  that  f o r any x n o t i n A i s e q u i v a l e n t t o  21 t h a t "^(R/A) = 0 ,  the l a t t e r ,  7Jj = Tp by P r o p o s i t i o n  in  turn,  i s equivalent  to  4.?«  (2) f o l l o w s from (1) by t a k i n g x = 1.  In  the  rest  of  this  section,  we study the r i n g o f  q u o t i e n t s w i t h r e s p e c t t o a PKF. L e t t = ^ Q  i s the  be a PKF,  ( f a i t h f u l l y ) t - i n j e c t i v e supporting  L e t E =• I (Q) endomorphism  be t h e i n j e c t i v e h u l l o f f^E^E]  ring  where  module o f t .  Q. . A l s o  denote  the  o f 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 t h e p r e v i o u s  notations,  (1) T i s a l o c a l ring.* (2)  The  J a c o b s on  radical  of  T  i s J (T)  = {.f  in  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  ring.  (5) Q i s a T-submodule o f E.  Proof.  (1)  E i s indecomposable i n j e c t i v e , t h e r e f o r e T  i s l o c a l £18 Theorem 25.4]. (2) Suppose Qf = 0. Then f i s n o t a u n i t i n T. Hence is map  in  f  J (T) , as I i s l o c a l . C o n v e r s e l y , suppose Qf # 0. The 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 K e r ( f ) — 0,  because  Q i s e s s e n t i a l i n E.  Indecomposability  o f 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) G i v e n f i n T, d e f i n e extension  of  f:  $ (f) i n S t o  Qf)Q£~ —>Q. l  Ker(j9)  = ( f i n T|f i s z e r o on  on Q} =  J  (4)  Then  be  the  unique  Q-: T-r->S i s a r i n g  Qf)Q£- } 1  = {f i n T|f i s  (T) . , T h i s i s an immediate consequence o f ( 3 ) .  (5) L e t f be i n T. Be have t o show t h a t Qf i s c o n t a i n e d in  Q.  The  map  8 (f) : Q~>Q  g; E—>E by i n j e c t i v i t y . QDQf *#  hence  -  is  defined  The  zero  map  on  Q.  f-g Thus  in  (3)  in T  extends t o  i s zero  Qf = Qg = Q  on  (f) i s  c o n t a i n e d i n Q.  A module H i s q u a s i - i n j e c t i v e of H, e v e r y  map  Eguivalently, as  right  in  {N,M ]  extends  to  a  map  i n itJ,Mj.  i s a submodule o f i t s i n j e c t i v e h u l l 1 ( H ) ,  £ I (H) ,1 (M) j-modules  Proposition  i f , f o r e v e r y submodule N  4.10(5)  means  £7,  Proposition  precisely  that  19.2(c) ]. Q  is  guasi-  injective.  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 t h e  following  equivalent conditions 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 t h e d i r e c t sum o f n c o p i e s o f H. (2) is  H g e n e r a t e s the c l a s s o f i n j e c t i v e H-modules.  That  t o s a y , e v e r y i n j e c t i v e module i s a homomorphic image o f  a d i r e c t sum of c o p i e s o f M.  Using t h e f a c t t h a t B i s p r o j e c t i v e ,  the equivalence of  map. zero  23 the two  previous  cofaithful  conditions  module  M  can  is  be  easily  necessarily  verified-  faithful, 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 IKF  A  f o r some  t , then t must be f a i t h f u l , because, i n t h i s c a s e , 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 a r e 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, Q i s i n j e c t i v e and R  Qs  i s f i n i t e l y g e n e r a t e d . T h e r e f o r e E = Q and B = £E ,E^.J i s  a  matrix  T  ring  over t h e d i v i s i o n r i n g T, i . e . , B i s s i m p l e  A r t i n i a a . But Q (B) — B as r i n g s [Example 3,5]- Hence t  Q^(B)  i s simple A r t i n i a n .  If  M  i s t - t o r s i o n - f r e e , 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 fi* = B / t ( B ) .  L e t F*  modules, c o n s i d e r e d of  torsion-free  be  a  module  the class  oyer  as B * - a o d u l e s . F» i s o b v i o u s l y  R  ,Q.  supporting T'«  a  class  B'-modules. T h e r e f o r e i t d e f i n e s an IKF t ' on  ,  module  ring  of a l l t - t o r s i o n - f r e e  on t h e c a t e g o r y M« of B -modules. I f t i s a PKF supporting  the  module ^Q, t h e n t« i s a PKF on M* w i t h Let  Q»  be  the  (faithfully)  fl  with  supporting  t»-injective  B*-module of t ' , and E* = I(Q*) i n M*. J,. S« = C ^ Q S ^ Q ' l and B« = f E y , E » . J T  Also l e t  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 s u p p o r t i n g module o f t . I f Q i s c o f a i t h f u l as B * modules, then Q (B) i s s i m p l e A r t i n i a n , t  Proof.  I t suffices  Q t ( ) = Qt'C *) fi  B  to  show:  (1)  Q» = K , Q  and  (2)  r i n g s . T h i s i s t h e case because, i f t h e s e  a s  statements a r e t r u e ,  Q« w i l l be  B»-cofaithful  and  Q '(B*) t  w i l l be s i m p l e A r t i n i a n by P r o p o s i t i o n 4.12. (1)  because R»Q i s ( f a i t h f u l l y )  i s clear  t*-injective  and i s a s u p p o r t i n g module o f the PKF t * on H«. To see ( 2 ) , we observe module  M  i s naturally  It^M) =  I(g'H)  that  every  t-torsion-free  B-  a t * - t o r s i o n - f r e e B*-module. Hence  and Q (M) ~ Q '(15), as B- and t  t  B*-modules.  In  p a r t i c u l a r , Q (B) ^ Q ( B / t (fi)) ^ Q .('B ) a s B and B*-modules. r  t  Using  t  t  the f a i t h f u l  straightforward to  t-injectivity  verify  that  Q «(fi*) = Q (S) i s i n f a c t a r i n g t  t  of  Q (B), t  the p r e v i o u s  i t is  isomorphism  isomorphism.  4.14. Example. Suppose fi i s l e f t N o e t h e r i a n and P i s a prime ideal.. Let  t = l p be  the  PKF  in  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 ( B ) i s s i m p l e A r t i n i a n . I n p a r t i c u l a r , t  left  Noetherian  prime  ring  has  a  simple  a  Artinian left  c l a s s i c a l r i n g of quotients.  P r o o f . From t h e proof of P r o p o s i t i o n 4.8, we B / P > — > Q n,  where  see  that  Q i s the t - i n j e c t i v e s u p p o r t i n g module o f  t . I f t ( B ) = P, then Qt(B) i s s i m p l e A r t i n i a n by P r o p o s i t i o n  25 4.  13.  I n p a r t i c u l a r , take P = 0 i n a ring.  Then  t = max  by  the  left  Noetherian  prime  d e f i n i t i o n of t , and max  f Example 3.5(2) ]- T h e r e f o r e Q j (fi) i s s i m p l e  = cl  Artinian.  c  S e c t i o n 5. Quasi-prime K e r n e l F u n c t o r s  In t h i s s e c t i o n we study a two  sided  analog  of  PKF,  namely, q u a s i - p r i m e k e r n e l f u n c t o r s . The main r e s u l t i s t h a t in  a  l e f t Noetherian r i n g every quasi-prime k e r n e l  i s Cp f o r some prime i d e a l  P.  L e t t be an I K F , r e c a l l t h a t t» less  functor  is  the  greatest  than o r e q u a l t o t , p r o v i d e d such a SKF e x i s t s . On  SKF the  o t h e r 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 t o p o l o g y , the c o r r e s p o n d i n g I K F symmetric then  and  t° = t * .  particular,  is The  that  Noetherian r i n g  denoted  exists  Proposition for  every  ideals.  implies,  IKF  t  in  in a left  (see [ 28 ]) .  5.1. P r o p o s i t i o n . L e t t be an IKF and I be dense  then  by t * . O b v i o u s l y , i f t * e x i s t s ,  following t*  is  the  set  I f I has a base o f i d e a l s which a r e  of  t-  finitely  generated as l e f t i d e a l s , then t * e x i s t s .  P r o o f . He have t o show D*  satisfies  (1) : and  (2)  of  26  Definition ('I)  1,5. 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, t h e n £D:xJ c o n t a i n s the same i d e a l I . (2) ideal  Suppose A i s a  I,  B  is  left  ideal  containing  a  t-dense  a l e f t i d e a l such t h a t [ B : a | c o n t a i n s a t -  dense i d e a l f o r e v e r y a i n A. We have t o show B  contains  a  t-dense i d e a l . We can assume t h a t I i s f i n i t e l y g e n e r a t e d as H  a  left  ideals  I = ZL  fia-.  Suppose I £ i s a t-dense i d e a l *='  is  11  a  t-  h  dense  i d e a l . L e t J = 21  Ka?. Then o b v i o u s l y J i s c o n t a i n e d  i n B. J i s an i d e a l because , f o r x i n 8, i s c o n t a i n e d i n J . J i s t-dense because  ( K a ) x = KCJTx^a;) k  I/J  is  t-torsion. ,  T h e r e f o r e 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 .  Let  t  be  an  SKF,  an E-bimodule S i s a  q u a s i - s u p p o r t of t i f : (1) S i s t - t o r s i o n - f r e e , and any aon^zero sub-bimodule S' o f S, S/S  5.3. D e f i n i t i o n . , An  t  (2)  for  i s t^torsion.  SKF t i s a q u a s i - p r i m e k e r n e l f u n c t o r ,  a b b r e v i a t e d a s QPKF, i f t h e r e i s a q u a s i - s u p p o r t S of t such that ^ *  e x i s t s and e q u a l s t .  The s t u d y of I K F and QPKF was i n i t i a t e d by Murdoch Oystaeyen*  In  variety theory Noetherian  £28], was  a  non-commutative  defined  on  the  and  v e r s i o n of a f f i n e  spectrum  of  a  left  prime r i n g 8, where t h e s t a l k at each P i n SpecB  i s Qg, (R). I t i s known t h a t i n a l e f t N o e t h e r i a n r i n g  every  27 Op f o r a  prime  ideal  P  is 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.UThen  P r o p o s i t i o n . L e t t be a QPKF and t ( 1 ) , t ( 2 )  be  SKF*s-  t e g u a l s t h e meet o f t {1) and t (2) i m p l i e s t = t (1) o r  t = t (2) .  Proof. L e t t = Then  where s i s a  guasi-support  = t (1.) (S) f) t (2) (S) . T h e r e f o r e t ( 1 ) (S) = 0 o r  0 = t(S)  t ( 2 ) (S) -= 0. I f t ( 1 ) <S) = 0, then t{1) < T = t  $m  Hence t ( 1 ) < ^ *  and t ( 1 ) = t . S i m i l a r l y , i f t (2) (S) = 0, then t (2) = t .  Denote t h e s e t o f t - c r i t i c a l i d e a l s C(t).  of t .  Due  to  the  fact  that  (excluding  t-dense  B)  ideals  by are  m u l t i p l i c a t i v e l y c l o s e d , C ( t ) c o n s i s t s o n l y o f prime i d e a l s . The f o l l o w i n g P r o p o s i t i o n i s c l e a r .  5.5.  P r o p o s i t i o n . L e t B be l e f t N o e t h e r i a n and s  and  t  be  SKF's. I f C(s) = C ( t ) , t h e n s = t .  Suppose  S  i s an B-module and K i s a r i g h t i d e a l o f B.  D e f i n e r (K) = £x submodule. bimodule.  If S  i n S| Kx - 0 ) . I n i s an  B-bimodule,  general, then  r (K)  r ( K ) i s a sub-  D e f i n e G(S) = (c i n B| r (cB) = O j , where S  B-module. C l e a r l y G{S) = G ( I ( S ) ) .  is a  i s an  28  5.6. P r o p o s i t i o n .  Suppose fi i s l e f t N o e t h e r i a n . An SKF t i s  a QPKF i f and o n l y i f t = Op f o r some prime i d e a l P.  P r o o f . I t has been noted t h a t each cTp i s a QPKF,  where  P i s a prime i d e a l . Let G(S)  t = f* s  be a QPKF w i t h g u a s i - s u p p o r t S. Then P = fi-  - (x i n S| r(xfi) # Oj i s an i d e a l , t h a n k s t o t h e f a c t  t h a t S i s a g u a s i - s u p p o r t . S i n c e fi i s l e f t N o e t h e r i a n , e v e r y i d e a l properly containing P contains  an  element  of C ( P ) .  C o n s e q u e n t l y , C(<Tp) = ( P j . By P r o p o s i t i o n 5 . 5 , i t remains t o show C ( t ) ••= { P j . The l a s t c o n d i t i o n i s e q u i v a l e n t t o t h a t an ideal  A i s t-dense i f and o n l y i f t h e i n t e r s e c t i o n o f A and  G(S) i s not empty. Suppose c i s i n t h e i n t e r s e c t i o n o f A and  G (S) ,  i s n o t empty. S i n c e £fi/ficfi^I(S) j = 0, Bcfi - i s 7^-dense,  which hence  t-dense. T h e r e f o r e A i s t-dense. C o n v e r s e l y , suppose A i s t dense,  hence  1$-dense. Then i t f o l l o w s from £fi/A,I (S) J = 0 n  t h a t r{A) ••= 0. L e t A = |£ B a left  ideal*  Then  be f i n i t e l y  f)?. r ( a j f i ) = 0.  bimodule o f S i s an e s s e n t i a l for  t  But  submodule,  generated  as  a  each non-zero subhence  r(ajtfi) • •= 0  some k. C o n s e q u e n t l y , a^ i s i n t h e i n t e r s e c t i o n o f A and  3  G(S). 5.7. P r o p o s i t i o n . , Suppose fi i s l e f t N o e t h e r i a n and t i s a PKF, t h e n t * (which does e x i s t ) i s a QPKF.  P r o o f . 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).  Tfi ^ %  P = £A:B] i s a prime i d e a l and  (see P r o p o s i t i o n 4 . 7 ) . By  s u f f i c e s t o show t h a t Op <  Example  3-8(2),  - 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 £B/J,I (R/A) } # 0.  i t  I t follows  that  there  dense, then  i s some x i n 8,  which i s not i n A, such t h a t J x i s c o n t a i n e d i n A. But A i s a  prime  l e f t i d e a l , hence J i s c o n t a i n e d i n A. T h e r e f o r e J  i s c o n t a i n e d i n P and J i s n o t Op-dense.„  We i n t r o d u c e quasi-supports  the f o l l o w i n g  of  a  QPKF  technique  and  their  t o study v  rings  of  the  oimodule  endomorphisms. L e t K be a commutative s u b r i n g identity  element)  contained  of  R  (with  i n the centre i s , by  R.  The  of  B  fi = E ^ f i o ,  B°  i s t h e o p p o s i t e r i n g of fi, i . e . , B°  where  K  of  enveloping algebra e  over  t h e same  definition,  = [x<»|x i s i n E} w i t h a« • b° = (a+b) °, aObP - (ba) o, ..H - i s a ring with multiplication  d e f i n e d by (a8b°)(c®d°) = ac8b°do  = ac®(db)o. Let  B  K  be t h e c a t e g o r y whose o b j e c t s a r e 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 c a t e g o r y a r e j u s t maps o f 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. L e t t h e c a t e g o r y o f B -modules be denoted by H*. Then  M»  i s naturally equivalent to B . K  From t h i s p o i n t on, we s h a l l f i x K t o be the s u b r i n g o f B  t h a t i s generated  by 1. Thus B i s t h e c a t e g o r y o f a l l BK  30 Suppose t = T *  bimodules.  T  i s a QPKF on M with, g u a s i - s u p p o r t  T and S i s a n o t h e r g u a s i - s u p p o r t o f t . L e t tj  be t h e IKF on  M' ( c a t e g o r y of B -modules) c o g e n e r a t e d by T, c o n s i d e r e d  as  an B -module. 5.8. P r o p o s i t i o n . S i t h t h e p r e v i o u s 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 o r S. (3)  (S) i s e i t h e r 0 o r S.  Proof.  (1)  F o r any B-submodule N o f t ^ M ) , C  Hence f o r any B*-submodule N  'Ey(H) #  of  N  # 2 T  £N,T ] e = 0 .  This  ff  means t h a t tj{IA) , as an B -module, i s T y - t o r s i o n . e  °-  =  R  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 also  tj-torsion-free.  Therefore  ^ ( S ) = S. T  (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 t h e n o t a t i o n s o f 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 * e x i s t s , then ^j* i s s  a QPKF w i t h g u a s i - s u p p o r t S. (2)  I f S i s T ' - t o r s i o n - f r e e , then T  ^* i s a PKF on H  w i t h s u p p o r t i n g module S. ,  Proof.  ( 1 ) tp(S) = 0 means  1j< %.  Therefore S i s a guasi-support of  Hence  ^7*  <  and VTj^* i s a QPKF.  1  31 (2)  I t follows  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 -submodule S* o f S, S/S* i s 2y»-torsion. Hence  S  is  by  e  a  /  supporting  definition,  the  module  previous  s u p p o r t i n g module o f  Suppose i s naturally  of  argument  - Hence  t = Zg*  T j . , Since  (T) - 0  shows  that  T  is  a  ^ * i s a PKF.  i s a QPKF w i t h q u a s i - s u p p o r t S. Q ( S )  /  t  an B-bimodule because t h e r i g h t  multiplication  on S by an element o f B extends u n i q u e l y t o a map o f l e f t  R-  modules  be  on  expressed  Q (S).  I t follows  t  that  every  QPKF  can  i n t h e 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  ring  of  bimodule  endomorphisms of a i s e q u a l t o f a , a j « = U. C o n s i d e r a as  a  R  r i g h t 0-module. The r i n g o f 0-endomorphisms [M./.ajyOf a w i l l be  called  the  quasi-bicommutator  of  H. , The  bicommutator o f a i s e q u a l t o t h e bicommutator  of  quasi-  the  B e  module a .  5.10. P r o p o s i t i o n .  QPKF  with  ( f a i t h f u l l y ) t - i n j e c t i v e q u a s i - s u p p o r t Q. L e t t*  = Ttf  and  = Qt'CQ)-  Denote  by  00  t =  fy*  a  Q*  Suppose  is  t h e B -module map OC: B — > B e  d e f i n e d by OC (a«b<>) = ab. C)  £ Q # Q J ^ e - [ Q S Q ' l / j e and they a r e d i v i s i o n  (2) I f C t - M t ( f i ) ) = t«(B*), and B / t ( B ) > — > Q n , n,  as  B - m o d u l e s , then t h e q u a s i - b i c o m m u t a t o r s e  are both s i m p l e  Artinian.  rings, f o r some  of Q and Q*  32 Proof. L e t 0 observe  = EQ.,Q]geaad  0' = [Q» ,Q» J  He  e.  R  first  t h a t Q i s a r i g h t U*-submodule o f Q':  Let f be a map o f non-zero  fi-bimodules  i n U'. S i n c e Q f l Q f  is a  - 1  sub-bimodule o f Q, t h e r e s t r i c t i o n o f f t p -.QflQf"-  1  extends u n i q u e l y t o a map $ (f) : Q—>Q o f l e f t B-modules  Q*  # (f) must be a map o f B-bimodules, because, f o r any r i n B, t h e map f : Q—>Q d e f i n e d by f ( x ) = # (f) ( x r ) - { $ (f) <x) ) r f  is  zero  r  on Q f l Q f , - 1  hence i s z e r o on Q„ .Secondly,  b e i n g a map o f B -modules, extends €  g: Q»—>Q',«  uniguely  $(f),  t o an B*-map  map f - g i n U' i s z e r o on Q f ) Q f # hence i s - 1  z e r o on Q». T h e r e f o r e Qf = Qg = Q #{f) i s c o n t a i n e d  i n Q.  T h i s shows Q i s a IM-submodule o f Q*. Now  i s c l e a r because t h e map Q :  (1)  the p r e v i o u s paragraph  D*r—>U  defined i n  i s an isomorphism o f r i n g s .  To see {2), we n o t i c e t h a t , under t h e hypotheses,  Q» i s  B /t«(B )-cofaithful: e  e  fi*/t« <B ) — > B / t (B) > — > Q > — > Q « " .., e  B  M  Y L7#- P r o p o s i t i o n 19.15], Q» i s a f i n i t e d i m e n s i o n a l  right  v e c t o r space over t h e d i v i s i o n r i n g 0*. T h e r e f o r e the g u a s i bicommutator  of Q*  i s simple  Artiniaa.  By t h e p r e v i o u s  33  o b s e r v a t i o n , Q i s a subspace o f bicommutator Artinian.  of  Q = CQ,Q3  U  Q*.  Therefore  = [Q*Q1 . U  the  i s also  guasisimple  34 CHAPTER I I  LOCALIZATION OF GROUP RINGS  The c e n t r a l between  the  question of t h i s chapter  rings  of  quotients  i s the  relation  o f t h e group r i n g RG and  those o f t h e group r i n g RH, where H i s a normal subgroup G.  In t h i s a r e a , t h e r e a r e s t i l l  unsolved  of  g u e s t i o n s o f very  s i m p l e appearance (see S e c t i o n 8 ) . The c o n t r o l l e r o f an i d e a l  i s a  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 In  Section  familiar  6 we s h a l l i n v e s t i g a t e  some consequences o f t h e  t h e c o n t r o l l e r o f a base o f a t o p o l o g y  of o u r r e s u l t s ,  though . developed  in  ( s e e , f o r example, £30]).  e x i s t e n c e o f a proper c o n t r o l l e r o f a t o p o l o g y . 7,  notion  out  of  In  Section  i s s t u d i e d . Some an  independent  c o n t e x t , do c o i n c i d e w i t h 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 t h e  t h e o r i e s of c o n t r o l l i n g subgroups.  S e c t i o n 6. The  Controller  L e t G be a group and B be a r i n g . By t h e group r i n g we £rg  mean with  the r i n g c o n s i s t i n g r  i n R,  g  of a l l t h e f i n i t e f o r m a l sums  i n G.  The  componentwise, and t h e m u l t i p l i c a t i o n ( 21 r„ x) ( X  Sj  BG  y) = SI ( 21 r s ) . z  addition  i s defined  i s d e f i n e d byz  35  Let  H  be  a  subgroup  of  G.  Suppose  X  is  a  left  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 u n i o n of  xH,  where x i s i n X. We  is  a subring by  of BG,  always assume t h a t  and  Then SH  BG i s a f r e e r i g h t BH-module  generated  X. Q:  There i s a map $  1 i s i n X-  u  H  ( Sl r„ g) = ,21 r#h. 3&i  ten  3  h  a * s are i n BH, x  then  BG—>BH o f RH-bimodules, d e f i n e d  E q u i v a l e n t l y , i f a = 21 xa«,  where t h e  *6X  9^ (a) = a^. Moreover, a  x  ~  by  By ( x ~ a ) . l  Suppose I i s a l e f t i d e a l o f BG., I t i s c l e a r t h a t (1)  (2)  The  > I  & (I) H  0 BH.  (BG) ( B {I)) H  following  > I >  Proposition  e q u a l i t i e s o c c u r i n (1) and 6.1.  DBH).  describes  the  case  (1)  $k  CD  = I 0  (BG) { fa (I))  (3)  (BG) (I fl BH) &H  following  equivalent:  (2)  m  where  (2).  P r o p o s i t i o n . With the p r e v i o u s n o t a t i o n s , the  a s s e r t i o n s are  when  (RG) (I  BH. =  = I.  (I) * I  6\d).  (5)  I = 51 x  The  p r o o f i s easy and  hence o m i t t e d .  We  remark  that,  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  In t h i s case,  (5) can  be w r i t t e n  as  I  where x $ ^ ( I ) x - * i s a l e f t i d e a l o f BH,  G.  = SL~ (x fi„(I)x-»)x, f o r each  x.  36 6.2.  Definition.  If  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  left  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 e v e r y member of F.  L e t H < K be subgroups of G such t h a t H c o n t r o l s a i d e a l I , then K c o n t r o l s I . On are  subgroups  of  G  controlling  i n t e r s e c t i o n of fl and  H(F)  be  the  we  v e r s i o n of [ 2 , Lemma 2 . 3 ] ;  intersection  subgroups of G c o n t r o l l i n g F.  (2)  and  K  K c o n t r o l s I . By t h e s e o b s e r v a t i o n s  of  ideals  of  a l l subgroups  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  (1)  and  the  P r o p o s i t i o n . L e t F be a f a m i l y of l e f t  Let  fl  a l e f t i d e a l X, then  can prove a g e n e r a l i z e d  6.3.  the o t h e r hand, i f  left  of  all  RG. of  G  normal  Then:  N (F) c o n t r o l F.  a subgroup fl c o n t r o l s F i f and  normal subgroup K c o n t r o l s F i f and  only i f fl > H (F) ; a  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 o f l e f t i d e a l s of RG t h a t I g i s i n F f o r e v e r y I i n F and e v e r y g i n G, t h e n =  such  W(F)  N(F).  P r o o f . L e t fl - 8 ( F ) . I t s u f f i c e s t o show gHg-i c o n t r o l s F f o r any <9  -,M)  g i n G. =  9  T h i s a s s e r t i o n f o l l o w s from t h a t fe(g- ig)gl  1  ^  g  8 a9)9~ l H  *  g(ig)g-  1  =  i -  37  The normal subgroup N (F) i s c a l l e d t h e c o n t r o l l e r o f F. As an example, l e t B be commutative and F be t h e s e t of a l l i d e a l s o f BG which a r e 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 A *G < N (F) = W (F) < A G, where ^ G = fx i n G|£G?C(x) ]  says  i s f i n i t e ) i s the  F.C. Subgroup  of  G,  JL\ *G  and  is  the  t o r s i o n subgroup o f ^ G. The f o l l o w i n g i s a n o t h e r example:  6.5. Example.  L e t fl be  a subgroup o f G. The augmentation  ( l e f t ) i d e a l o f H i s the l e f t i d e a l cd^R of BG g e n e r a t e d 1-h, where h i s i n H: con  =  -V H.  By  H  BG(1-h) =  HZ  Proposition  oo^W. We  controls  =  COQH  7LZ.  by  x ( A) H) , where  6.1.(5), i t i s obvious that H  claim:  (1) H = W( O^H) . (2) I f fl i s n o r m a l , then H = H( (( W^H)  in=1,2,...J) .  W  To see ( 1 ) , we assume, on t h e c o n t r a r y , t h a t some  x  in  H  there  is  which i s not i n W = W(4)^.H). S i n c e 1-x i s i n  COqR, i t f o l l o w s t h a t 1 -  8u/(l-x)  is in  <50^.H.  This  is  a  contradiction. 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  for =  every  n=1,2,--..  By  induction  on  ( co^tt)"  n,  Z±Z x ( W H ) . T h e r e f o r e H c o n t r o l s ( c*s i\) . n  n  &  For t h e r e s t o f t h i s s e c t i o n we s h a l l s t u d y b r i e f l y t h e c o n t r o l l e r of a t o p o l o g y . We H, (F)  = H (F)  for  a  remark  that,  by  Proposition  t o p o l o g y F. Thus we c o n s i d e r t h e  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 t o p o l o g y o f l e f t i d e a l s of BG. H c o n t r o l s  G.  F  is  a  38  D e f i n e t h e i n d i c a t o r of ¥ t o 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 a r e i n t e r e s t e d i n t h e  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  notations  and  a s s u m p t i o n s , i f K(F) = G, then G/H i s a t o r s i o n group.  Proof.  Suppose,  such t h a t x  n  on t h e c o n t r a r y , t h e r e i s some x i n G  i s n o t i n H f o r every n=1,2,.... By a s s u m p t i o n ,  t h e r e i s a dense l e f t i d e a l I * BG such t h a t l x < I . L e t Y be a l e f t generated  by  H  and  transversal x,  of  <H,x>,  the  subgroup  i n G. Then { y x | y i s i n Y, n = an w  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 o f H i n G-, S i n c e BG(1-x) + 1 =  $H(1-X)  in  I  i s dense (hence c o n t r o l l e d by  i s i n BG (1-x)  such  that  H) ,  1  + I . Thus t h e r e e x i s t a i n BG and b  a (1-x)  • b  = 1-  Applying  t o both  s i d e s , we have  (*) for  ( Z x ' a ) (1-x) • £ x b l  (  i  = 1  some a.L i n Bfl, b j i n #/y(I) - B H f ) I . I n v o l v i n g o n l y f i n i t e sums, e q u a t i o n  (*) can be r e d u c e d  to: ( iE  x*a.) (1-x) • 2__ x ^ b i = x , 0<k<m, fc  or: (**)  ( a , + b . ) + 2 : x ' ^ - a ^ + b - ) + x-*M-ai) = x  f e  where a * = x-*ax. The either  c o e f f i c i e n t s of x 0  or  1,  are  1  on both  sides  invariant  of  under  (**) ,,  being  the a c t i o n  39 y "I  { )  ,  ~ x ( ) x - . Thus (**) 1  (***)  (a,tb,)* X  i s f u r t h e r reduced t o :  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 of  (***) , »e Zl  on both  sides  obtain: bi  = ^  x-" b x  = 1  ;  Thus 1 i s i n I - T h i s i s a c o n t r a d i c t i o n . 6.7.  Proposition.  Let  H  be  a  normal subgroup o f G which  c o n t r o l s a non-zero t o p o l o g y F. I n  each  of  the  following  c a s e s , K (F) •= G: (1) There e x i s t s a p r o p e r 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  is  non-zero means F#{BG}, o r , 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 , t h e n K(F)  > {x  in  Gl  Ix = 1} = G. (2)  Every dense l e f t i d e a l i s i n the f o r m I •= 2 Z x i * ,  where I * i s an i d e a l of BH. Thus {x  in  GiIx = I j  = G  for  every I i n F. (3) Since x  w  Let  x be i n G and I be a proper dense l e f t  = 1 f o r some i n t e g e r n, J = f l ; " I x  i d e a l s a t i s f y i n g Jx = J .  c  ideal.  i s a dense l e f t  40  Section  7.1.  7. The R i n g s Of Q u o t i e n t s . O f _ A Group B i n q  D e f i n i t i o n . A normal subgroup H o f G weakly c o n t r o l s  a  t o p o l o g y 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 family of l e f t i d e a l s of  BG, F* i s a f a m i l y o f l e f t i d e a l s o f BH, where H i s a normal subgroup o f G. Then the f o l l o w i n g a s s e r t i o n s a r e e q u i v a l e n t : (1)  F  is a  topology;  H  weakly  controls  F;  F*  - { 0H(») ID i s i n F j . (2)  F*  is a  topology;  f o r any g i n G and D* i n F*,  g-iD«g i s i n F ; F = {D|D i s a l e f t i d e a l of  BG  containing  D* i n F»j, where X i s a l e f t  transversal  f  51  xD*  for  H i n G.  f o r some  (3) F' i s a t o p o l o g y ; f o r any x i n x-»D*x  is  i n F«; F•=  and  D»  {D|D i s a l e f t i d e a l o f RG  2Z xD« f o r some D" i n F * l , where X i s a **y for  X  left  in  F*,  containing transversal  H i n G.  Proof.  To  show  (2) and (3) a r e e g u i v a l e n t ,  (3) and w r i t e g = x h , where x i s i n X and h i s  we assume  in  H.  Then  g-iD'g = x-*D*xh = £x-*D*x:h- J i s i n J  To  show  ( 1 ) i m p l i e s (2) « we suppose ( 1 ) h o l d s . I t i s  obvious that F equals the s e t containing be i n F*  = Qfl  some with  (X-*DX)  =  of  a l l left  ideals  xD», where D* i s i n F«. L e t D' = D  i n F.  Then  f o r any  x  in  G,  of  BG  $«(D) x-tDfx  (Dx) i s in F', because Dx i s i n F. Thus i t  41 remains t o show t h a t F* i s a c t u a l l y a Let  a»  be i n BH and A* =  He show B a * < A* f o r some B ,  such  that  Ba«  < A-  1  topology-  # (A) be i n F* w i t h A i n F. H  i n F'.. There i s some  Hence  ^(B)a»  <  0^ (B) i s i n F». Next l e t A» and B« be such  that  C\,  B»{k)  left  B  in F  = A«, where  ideals  of  BH  A* i s i n F* and, f o r any a* i n A*, t h e r e i s some  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 . B y assumption, - "ZZ xA*  i s i n F., I t s u f f i c e s  = S I xa„ i n A, where a*, i s i n A*, where  A  t o show t h a t , f o r any a there  is D  =  S I xD»,  D* i s i n F*, such t h a t Da < B- By what we have proved  so f a r , D' -  O xrB»:a-:lx- i s such a l e f t a  To show (2)  implies  ( 1 ) , we  first  ideal. observe  that  H  c o n t r o l s t h e base (S~xD'|B* i s i n F } o f F. Suppose A i s i n 1  F and a = ZzZ x a i s i n BG. He have t o show t h e r e i s some B* x  F*  for  some A* i n F'« Then B» =  ideal-  such t h a t  { S I xB»)a < A. By assumption, A >  in  f)  x[k* i<-.,c]x-i  i s such  2Z xk* a  left  Suppose next t h a t A and B a r e l e f t i d e a l s of BG such  t h a t A i s i n F and i B : 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 D i n 1  *«-x Suppose A > HZ^ xk*, where A* i s i n F*. Then f o r any a*  F*.  i n A , t h e r e i s some 1  L e t D* = ST Louden  (B«,)a*.  EK,  i n F» such t h a t  { 27  Then D« i s i n F» and B >  has i n t r o d u c e d  left  ideals  of  B.  <  B-  ST xD«-  t h e n o t i o n of an S-qood  £24]: L e t f : B—>S be a map o f r i n g s . L e t P" be of  xB*.)n*  a  topology topology  D e f i n e F t o be t h e s e t o f a l l l e f t  42 i d e a l s D o f S such t h a t i r ( D ) i s i n F*. F» i s c a l l e d S-good 1  if F i s a  topology.  I n t h e c a s e o f f : HH—>BG, where H i s a normal subgroup o f G, t h e F so d e f i n e d Proposition  coincides  with  that  described  7 . 2 ( 2 ) , Thus the a s s e r t i o n t h a t  in  (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 w i t h p a r t o f [24>, Theorem 3.5 ]. For t h e r e m a i n i n g p a r t o f t h i s s e c t i o n , we suppose t h e conditions  (1)  to  (3)  of  P r o p o s i t i o n 7.2 a r e s a t i s f i e d .  Denote t h e I K F * s c o r r e s p o n d i n g t o F and respectively.  Let  BG®^  = BG« _ RW  F*  be  by  the  the  every  an  may  be  considered  as  and  tensor  f u n c t o r from BH-modules t o HG-modules. On BG-module  t  t»,  product  other  hand,  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  following  equalities  hold  i n the  c a t e g o r y o f 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. from  (1)  and  (2)  are straightforward.  (3) f o l l o w s  (2) by p u t t i n g M» = BH.  7.4. P r o p o s i t i o n . F o r any BG-module Q, Q i n j e c t i v e i f and o n l y i f Q i s f a i t h f u l l y  Proof.  i s faithfully, t t -injective. ,  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 o n l y  i f Q  i s t'-injective  under t h e  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  £ VQ3 RH ^ I BG8M* , Q 3 a s a b e l i a n groups., I n f a c t , t h e map M  f  i n £RG®H»,Q]^ c o r r e s p o n d i n g  t o f« i n X&'+QIRH  i  defined  s  by f (g®m) = g <f• (m)) , f o r g i n G and m i n M«. T h e r e f o r e , we  have  BG8D»  < E  i f  < BG, where D» i s i n F« and E i s i n F,  t h e n we can c o n s t r u c t t h e f o l l o w i n g diagram  C BG , Q ]  >CBG8D«, Q3  u-  Suppose Q i s t - i n j e c t i v e , then u i s e p i c f o r any D* i n F *.  Therefore  u* i s e p i c f o r any D* i n F«. T h i s shows Q i s  t•-injective. Conversely, uCii  i s epic  suppose Q i s t ' - i n j e c t i v e . We have t o show 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'. S i n c e u  i s epic,  1  u< >u< > 2  1  is  e p i c . L e t f be i n £ E , Q ] ^ . Then u< >f = u< >u< *>g f o r some g 2  in  £BG,Qj ., a/  o r u< >(f - u<*»g) 2  2  = 0. But Q i s t - t o r s i o n -  f r e e . T h e r e f o r e f - u<* >g = 0, and u< > i s e p i c . l  R e c a l l t h a t an IKF i s N o e t h e r i a n corresponding  localization  functor  i f and o n l y preserves  L e t t h e l o c a l i z a t i o n f u n c t o r s o f t and t» be  i f  the  d i r e c t sums.  denoted  by  Q  44  and Q', r e s p e c t i v e l y .  7.5.  Proposition.  (1)  F o r any BG-module M, Q (M) ^ Q« (H) as  BH-modules. (2) Q(f) = Q* ( f ) f o r any map f o f BG-modules.  P r o o f . By P r o p o s i t i o n s 7.3 and 7.4, Q{M) •t*.-injective  and  Q(M)/M*  is  faithfully  i s t«-torsion, where a * = M/t(M)  = M/t' (M>. T h i s p r o v e s ( 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 N o e t h e r i a n i f and  only  i f t' i s  I f t i s N o e t h e r i a n , i t f o l l o w s from  Proposition  Noetherian.  Proof. 7.2(2)  that  preserves then  t*  direct  © Q(Hc)  Noetherian.  sums. I f { M ) ;  i s faithfully  hence f a i t h f u l l y ® Q(Mj)  is  Conversely,  Q*  i s a f a m i l y of BG-modules,  t ' - i n j e c t i v e by P r o p o s i t i o n  t - i n j e c t i v e by P r o p o s i t i o n  contains  suppose  7.4.  7.5,  Moreover,  ( ® M ; ) / t ( © B ; ) a s a t»-dense, hence t -  dense, submodule. T h e r e f o r e  © Q (M^) .= Q ( 0 H ; ) . .  7.7. P r o p o s i t i o n . F o r any BH-module  M»  and  any  x  in  G,  Q» (xM») ~ xQ» (M») as BH-modules.  Proof. then  If  xN*  is  N* i s an a r b i t r a r y considered  a»xn»=x(x-*a x)n», t  as  BH-module and x i s i n G, an  BH-module  via  a» i n BH, n« i n N». O b v i o u s l y xQMM') i s  t - t o r s i o n - f r e e containing  (xM») / t * (xH») ^ x (M'/t-* f.H*)) as a  1  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 ) , c o m p l e t e s the p r o o f : ,  _ 1  £BH,xQ« (H».) j ^ —  (RH  7.8.  ,Q«(M») ]  •  >£D»  , xQ• (H*) j  f i A  ,  >fx-lD«X,Q» ( H , ) 3 f t H  w  P r o p o s i t i o n . L e t M' be an BH-module., (1)  There  is a  unique  monomorphism  g  making  the  f o l l o w i n g diagram o f BH-modules commutative;  Q* (M*) >  H«  >-  —-q  > Q(BG«M«)  —£-  >  BG®M*  where f(m») •= 1«m* f o r any m* i n fi', and u and  u  1  a r e 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 o f Q(BG»H«) . (3) I f e i t h e r  ( i ) £G:H J  i s finite  or  ( i i ) t« i s  N o e t h e r i a n , then (EG) q (Q* (ft*) ) "= Q (BG8H*) .  Proof. Proposition  (1)  This  i s an  immediate  consequence  7.5, because Q(BG®M*) = Q» (BG«M«) i s  of  faithfully  t * - i n j e c t i v e . S i n c e f i s mono and g = Q* ( f ) , q i s a l s o mono.  (2) =  Since  q  i s mono,  © x Q * ( H * ) = RG » Q*  because  i t  (HG) g (Q« (M»)) = <5> xq(Q» (M*) )  . T h i s i s an  contains  the  essential  essential  submodule submodule  (RG«H»)/t» (£G«fl ) o f 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 either  (i) or  ( i i ) h o l d s , .fiGq (Q*  ©xg(Q*(H»)) i s a l s o t»-injective. Hence i t i s a  summand  o f Q (RG®M*). I t f o l l o w s from  (2) t h a t  (4'<))'  direct  (BG)g(Q* («•.).)  = Q(RG8M*) .  7.9. P r o p o s i t i o n . . (1), There i s a makinq  the f o l l o w i n g  diagram  unique  ring  embedding  i n the category  q  of rings  commutative:  Q* (RH) u*  q  T  > Q (RG) T U  RH  f  >  RG  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)  r i n g s t r u c t u r e o f RG ® Q*(8H) = xySy-ia'yb*. (3)  a s r i n g s , where t h e  i s defined  u (RG) g (Q* (RH)) i s a s u b r i n g  I f either  ( i ) «.G:H]  i s finite  by  (x«a*)(y®b')  of Q (RG) . or  (ii) t' i s  N o e t h e r i a n , then u (RG) q (Q* (RH) ) = Q (RG) .  Proof.  (1) R e p l a c i n g  M* by RH i n P r o p o s i t i o n 7.8(1) , we  47 obtain  an  embedding  satisfying Q (BH)  = uf.  of  BH-modules  Using t h e f a i t h f u l t ' - i n j e c t i v i t y of  and Q (BG) , i t i s easy t o show t h a t q i s a  J  On  qu»  q: Q »(BH)—>Q(RG)  the o t h e r  hand,  any r i n g  ring  map.  map q must be a map of BH-  modules. T h e r e f o r e q i s unique. (2) By  Proposition  = BG 8 Q* (RH)  is a  u(BG)Q'(BH)  7.8(2),  submodule  = (RG). Q* (BH)  o f t h e RG-module Q(RG). The  r i n g s t r u c t u r e on BG « Q (BH) c o i n c i d e s w i t h t h a t  o f Q (RG)  1  on  the  t-dense  submodule  RG-  Therefore  RG 8 Q' (RH)  = u(BG)Q'(RH) a s r i n g s and i s a s u b r i n g o f 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. right  Proposition.  (1) Q i s r i g h t e x a c t i f and o n l y i f Q» i s  exact. (2) t i s p e r f e c t i f and o n l y  Proof.  i f t» i s p e r f e c t .  (1) By P r o p o s i t i o n 7.5 ( 2 ) ,  Q*  i s right  exact  i m p l i e s Q i s r i g h t e x a c t . C o n v e r s e l y , we show every D* i n F» i s t ' - p r o j e c t i v e , provided Let f and  N*  : H«—»H»  every D i n F i s t - p r o j e c t i v e .  and g«: D » — >H«  are t*-torsion-free,  projectivity  of  2 1 xD»  D'  be RH-maps, where M»  being  1  By  t-  = BG 8 D«, t h e r e i s E i n F and an  RG-map h: E—>BG 8 H* making t h e f o l l o w i n g modules commutative:  in F .  diagram  o f BG-  48  —>BG 8 D  f  »BG  BG8g« v 8 N*  By P r o p o s i t i o n 7.2(2), we can assume E =  21 x E  BG 8 M». —  in  j?'- Define  BG8f •  h*: E»—>J3*  by  h»  f  f o r some E*  = ph, where  p i s the  © xS'—->M». Then h* i s an BH-map and  p r o j e c t i o n p: BG8M•  the f o l l o w i n g diagram i s commutative;  E» >-  •> D<  h«  g*  M»  Therefore  - »  N  1  D• i s t»-projective.  (2) f o l l o w s from (1). and P r o p o s i t i o n 7.6,  Section  8. The Maximal Ring Of Quotients ftnd The C l a s s i c a l  Binq Of Q u o t i e n t s  Let  G  be  a  group  andflbe a normal subgroup. I t i s  s t i l l unknown what c o n d i t i o n on G and that  Q J»A  X  Q (BG) cl  MAX  e x i s t , what c o n d i t i o n  = BG ® Q (BH). Be s h a l l  i n the c o n t e x t  i s equivalent  BG ® Q ( B H ) , and, i f the c l a s s i c a l  (BG)  of g u o t i e n t s  H  i t  o f S e c t i o n 7.,  i s equivalent  to  to  rings that  d i s p l a y some known r e s u l t s  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 o f G. Then: (1)  Q (fiH)  i s uniquely  m A y  embedded i n t o Q  w 4 y  ( B G ) as a  s u b r i n g such t h a t t h e f o l l o w i n g diagram i s commutative:  > Q  (BG);  M A X  T U  BH  where  u«,  u  u(BG]q (Q  and  f  are  the  natural  ~ BG 0 Q^yCBH)  (BH))  m K l l  BG  and  maps. is  Moreover,  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 t h e p r e v i o u s embedding.  Proof. = max^ By  (1)  F* = t h e Lambek t o p o l o g y  i n P r o p o s i t i o n 7.2- F» i s G - i n v a r i a n t  Proposition  corresponding embedded  Put  7.2(2), IKF  into  we o b t a i n a t o p o l o g y Proposition  Q (BG)  as  a s u b r i n g and BG ® Q  on BG £9, Theorem 1 j . T h e r e f o r e , a subring  of Q  m A x  Q  (BH)  rtAX  m M t  t  (BH)  i n t h e Lambek  by P r o p o s i t i o n 2.5,  unique,  commutative:  we  is isa  topology Q*(BG)  (BG). Thus t h e embedding g i s o b t a i n e d .  The c o m m u t a t i v i t y o f the diagram i s o b v i o u s . To see is  p.528 ].  F of BG, w i t h  By  s u b r i n g o f Q (BG). But F i s c o n t a i n e d  is  £24,  t.  t  7.9,  on BH and t»  observe  that  t i e following  that  diagram  q is  50 BG ® Q *x(SH)  BG«g  w  >  (RG)  QmAM  I  RG  and  that  (RG 8 Q  m  (BH))/RG  M  is  t-torsion,  hence  raax^-  torsion. (2)  If  [G:Hj  P r o p o s i t i o n 7.9(3) Theorem  5  Therefore  and  i s finite,  that  Lemma  Q (BG) •= Q t  m M C  Be  BG 8 Q 10j, F  (BH)  i t follows = Q (BG).  = Lambek  concerning  the  one may c o n s u l t [ 2 5 , C o r o l l a r i e s  now  turn  from  By  t  [ 25,  topology  of  BG.  maximal  ring  of  (RG).  For f u r t h e r r e s u l t s quotients,  then  to  13, 14].  the c l a s s i c a l r i n g of q u o t i e n t s . The  f o l l o w i n g r e s u l t s a r e e s s e n t i a l l y due t o H e r s t e i n and S m a l l :  8.2. P r o p o s i t i o n , i 161 L e t B be a commutative r i n g , G  be  a  group and J G be i t s F.C. Subgroup. (1) Ore  If  G/AG  i s l o c a l l y f i n i t e , then C (BG)  i s a left  s e t (see Example 3.2 and S e c t i o n 9 f o r the d e f i n i t i o n s ) .  Hence Q \ (BG) c  exists.  (2) I f H i s a normal subgroup of G, c o n t a i n e d such  that  G/H i s l o c a l l y f i n i t e ,  in  4 G.,.  then, f o r any d i n C(BG),  t h e r e a r e d* i n C(BH) and a i n BG such t h a t ad = d*. (3) I f H i s a normal a b e l i a n subgroup of G/H  is  G  such  that  l o c a l l y f i n i t e , then, f o r any d i n C(RG), there a r e  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 subgroup H, H/4 also  left  G i s l o c a l l y f i n i t e , then f o r  B i s also l o c a l l y  Ore.  The  finite,  C(BH)  (1)  Q ^ (BH)  both  C (£G)  where H i s a normal subgroup of is  c  is  8.1.  P r o p o s i t i o n , suppose fi i s a r i n g , and a r e l e f t Ore,  C(fiH)  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 o f  q u o t i e n t s i s analogous t o P r o p o s i t i o n  8.3.  hence  any  uniquely  embedded  and  G.  i n t o Q ^ (BG)  as  a  s u b r i n g such t h a t the f o l l o w i n g diagram i s commutative:  Q  (BH)  4  g  > Q  cl  (BG) ju  BH  where u», u (BG)q (Q  ct  (2)  and  f  (BH)) "  is  G, G/H  are 6 .Q  RG  Suppose  (i) H < A G/H  u  f  cl  the (BH)  >  BG  natural and  maps.  Furthermore,  i s a s u b r i n g of Q  (fiG) .  cl  f u r t h e r t h a t B i s commutative, and i s l o c a l l y f i n i t e , or  (ii) H  is  l o c a l l y f i n i t e . Then fiG ® Q i(»H) = Q j (BG) c  £  either  abelian> v i a the  p r e v i o u s embedding.  Proof. Following let  t» = cl£H  contained  in  < BG ® Q i (BH) c  to  the p r o o f o f  obtain  C(BG),  an  Proposition  IKF  t < cl  t on BG. .  8.1(1),  S i n c e C (BH)  Therefore  < Q (BG) < Q ^ (fiG) , and t  we  the o t h e r  Q  is  ^(RH)  assertions  f o l l o w as i n the p r o o f 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  cl  (BH)  = Q (BG) t  because  52 c l  M  i s Noetherian. (2)  Under  the  assumptions,  8.2(2) and ( 3 ) . T h e r e f o r e EG ® Q  tl  t  =  cl  R (  r.  fay P r o p o s i t i o n  (BH) = Q (BG) = Q j (BG) . C  c  53  CHAPTEB I I I  GLOBAL DIMENSION OF_GBO0'P BINGS  The aim o f t h i s c h a p t e r i s to global  dimension  of  the  investigate  the  (left)  group r i n g BG, where e i t h e r B i s  commutative o r G i s a l o c a l l y  M-group.  In  particular,  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 , t h e f i n i t e n e s s o f l.gl.d.B  imposes a f i n i t e upper bound f o r l . g l . d . B G .  Historically, continuation generated  our  result  may  be  considered  a  o f H i l b e r t * s Syzygy Theorem; i f G i s a f i n i t e l y  torsion  free  abelian  group,  then  l.gl.d.BG  = l . g l . d . B • rank (G). Along t h i s l i n e o f approach, has  as  proved  Balcerzyk  t h a t 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 g e n e r a t e d , o f rank n, and  Z  is  the r i n g of i n t e g e r s £3, Theorem J .  Section  In  9.  Background  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 t h e o r y and h o m o l o g i c a l a l g e b r a .  These r e s u l t s can  be found i n £ 4 ] , £6J, £15], £ 1 7 ] , £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 (*) : of  1 = subgroups  G< 0  of  G, < G,  < 6n = G such  that  G i-t  is  normal  in  G -, t  1=1,2,...,n.  The  groups G;/G{-/ , i = 1 , . . . , n , are c a l l e d  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  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 Gi/Gt-i  each  the  (*)  is  invariant  i s c o n t a i n e d i n t h e c e n t r e of G / G ,  G, and  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  series  and  length  of  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 (*)  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 this  case,  the  upper c e n t r a l s e r i e s and  s e r i e s o f G a r e b o t h f i n i t e and common  length  a r e of  i f i t has  factors.  c a s e , the d e r i v e d  this  A l l n i l p o t e n t g r o u p s are A  group  invariant has  series.  the l o w e r c e n t r a l  equal  length.  G  is  a normal s e r i e s  with  This G.  abelian  s e r i e s of G i s f i n i t e .  solvable.  called  supersolvable  if  i t  has  an  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  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 ,  M—group  In  i s c a l l e d the c l a s s o f the n i l p o t e n t group  A group i s s o l v a b l e In  a central  if  it  has  a  G  is  normal s e r i e s w i t h f i n i t e or  an  cyclic  f a c t o r s . O b v i o u s l y , e v e r y s u p e r s o l v a b l e group i s p o l y c y c l i c , and  e v e r y p o l y c y c l i c group i s an A group G i s s a i d t o have  M-group. maximal c o n d i t i o n  subgroup  i s f i n i t e l y generated. E q u i v a l e n t l y ,  condition  i f every s e t of subgroups has  H—groups  have maximal c o n d i t i o n £33,  p o l y c y c l i c i f and  only i f G  is  a  i f  G has  maximal  every maximal  element.  7. 14]. T h e r e f o r e , G i s  solvable  and  has  maximal  condition. A group G i s l o c a l l y f i n i t e i f e v e r y f i n i t e l y  generated  55 subgroup  o f 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 a  normal  nilpotent  by-finite-properties follows  from  subgroup are  of  defined  [ 3 3 , 7.1.10]  in  finite a  index.  similar  has Other  way.  i t  t h a t M—groups, s u p e r s o l v a b l e  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 a r e  equivalent  terms. Suppose G i s f i n i t e l y g e n e r a t e d n i l p o t e n t , , central  series  1=Z°SZ*<.,,$Z =G. n  c o n d i t i o n , each f a c t o r Z^/Z^-i Lemma 1 . 7 ] group  is  local  The  lower  Then, due to the c e n t r a l finitely  generated  [15,  and a b e l i a n . Thus a f i n i t e l y g e n e r a t e d n i l p o t e n t supersolvable*  supersolvability  equivalent  is  with  Consequently, and  local  local  nilpotency,  polycyclicality  are  t o one a n o t h e r . preceeding  discussion  is  summarized  in  the  following diagram finitely generated nilpotent  abelian  li  nilpotent  supersolvable  li  locally nilpotent  polycyclic  %  polycyclic^^supersolvable by finite by finite M-group  locally polycyclic  nilpotent by f i n i t e  locally supersolvable  locally B-group  finite 4  locally finite  5 6  a normal s e r i e s with f i n i t e o r 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 . L e t G be a group, t h e n M-series,  provided  any  two  t h e y e x i s t , have e g u a l number o f 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 number  of  a  locally  M-series.,  The  Hirsh  M-group i s t h e supremum o f t h e H i r s h  numbers o f i t s f i n i t e l y generated subgroups*  We now t u r n t o h o m o l o g i c a l a l g e b r a . L e t S be a r i n g , an  S-module.  The n-th r i g h t d e r i v e d f u n c t o r of the f u n c t o r  Hom (_,B) from t h e c a t e g o r y o f S-modules t o t h e c a t e g o r y s  abelian  B  groups  is  denoted  by  Ext£(_,B).  of  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 d i m e n s i o n p. d.^ a (or  p.d.a,  i f S i s understood) of an S-module a i s t h e s m a l l e s t i n t e g e r n>0  f o r which  Ext"^-* (A,^.) = 0,  e x i s t s . I f t h e r e i s no infinite.  The  such  provided  integer  n,  such  an  then  integer  p.d. a 5  is  l e f t g l o b a l dimension 1.gl.d.S of the r i n g S  i s t h e supremum of  p.d. a, s  where  a  varies  over  a l l S-  modules.  It  i s well  equivalent:  (1)  known  that  the f o l l o w i n g c o n d i t i o n s are  p . d . s a = n. (2) t h e l e n g t h o f  the  shortest  57 projective  resolution  of  A  is  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. P r o p o s i t i o n . A>—>P—»B  £18,  Shifting  Theorem,  p.47 ]  Suppose  i s a s h o r t e x a c t sequence o f S-modules and p i s  projective.  Ext{? (A,_) ~ E x t " | M B > _ ) ,  Then  Conseguently,  p.d.A=0  n=1,2,*,..  i f and o n l y i f p.d.B=0 o r 1; p-d.A=n  i f and o n l y i f p.d.B=n*1, n=1,2,...; and p.d.A  is  infinite  i f and o n l y 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 . cohomological p.d.^B,  Let  £  be  a  ring,  and  d i m e n s i o n c*d.^G o f G w i t h  G  a group. The  respect  to  where S i s c o n s i d e r e d as an BG-module w i t h  R  i s  trivial  G-action.  9.5. P r o p o s i t i o n . L e t B be a r i n g , and fi be a subgroup o f  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 <  Proof.  l.gl.d.RG.  By £17, P r o p o s i t i o n 12.3],  Ext^(H,Hom {BG,B)) flH  = Extft {M,B) f o r any HH-module B. Hence <1) f o l l o w s . W  L e t A, B be a r b i t r a r y BH-modules. module  containing  isomorphism  in  A  the  i s an  H-BG^/^K  BG-  a s an B H - d i r e c t sumraand. A p p l y i n g t h e preceeding  paragraph,  we  see  E x t ^ ( _ , _ ) = 0 i m p l i e s E x t / J ( A , B ) = 0 . Heace (2) h o l d s . <  that  (  58 9.6. P r o p o s i t i o n . L e t B be a r i n g , and H be a subgroup o f a group G. C o n s i d e r t h e f o l l o w i n g (1)  For  any  conditions;  BG-module M, p.d.^M = 0 i m p l i e s p . d . ^ H  < n. (2) F o r any BG-module H, p.d.^H < p.d.^ M • a. H  (3) l . g l . d . B G < l . g l . d . B H + n. Conditions  (1) and (2) a r e e q u i v a l e n t ; and (2)  implies  (3).  Proof.  I t i s clear  have t o show  that  (2) i m p l i e s (1) and ( 3 ) . He  (1) i m p l i e s ( 2 ) .  Suppose (2) has been proved f o r a l l BG-modules M* p.d.^M*<k.  L e t p.d.^.H=k. C o n s t r u c t a s h o r t e x a c t sequence  N > — > P — » H of projective. (2)  is  with  By  valid  BG-modules,  where  P  i s BG-,  hence  BH-,  t h e S h i f t i n g Theorem, (2) h o l d s f o r M. Thus f o r a l l BG-modules  of  finite  projective  d i m e n s i o n ^ S u p p o s e p.d.^M i s i n f i n i t e . Then p . d . ^ H must be infinite, Shifting projective  otherwise, Theorem, but  by  there  of  repeatedly is  infinite  an  application  BG-module  BG-projective  which  of  the  is  BH-  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 . L e t B be a r i n g , and G be a f i n i t e group  of  order  cyclic  n. I f n i s n o t a u n i t i n 8, then c.d.^G i s  infinite.  P r o o f . I t i s w e l l known t h a t Ext *+ (B,B)=B/nB, f o r a l l 2  2  59 n>0  £14, p p . 3 9 - 4 0 j .  Ext2«*2{B,fi)#0,  Section  Let  If n  i s not a  unit  i n E,  then  f o r n>0. T h e r e f o r e c.d.^G i s i n f i n i t e . .  10. The Commutative  Case  B be a commutative r i n g , and G be a group. We s h a l l  prove t h a t gl.d.BG < g l . d . f i + c.d.^G.  E  being  commutative,  n a t u r a l r i g h t BG-module  every  left  structure  given  BG-module by  M has a  mg=g-*m, f o r  e v e r y element g o f G, and e v e r y element m of H. I n t h i s way, a  map  of  left  BG-modules i s n a t u r a l l y a map o f r i g h t BG-  modules. T h e r e f o r e 1.gl.d.BG = r . g l . d . B G . T h i s common of  value  t h e g l o b a l d i m e n s i o n 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 t h e BGmap  f : .BG&^A—>A,  d e f i n e d 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] L e t B  be  a  ring  (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 o n l y 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 , t h e n 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  projective,  60  then & i s i s o m o r p h i c  t o an .BG-direct summand o f BG« a. Hence R  a i s B G - p r o j e c t i v e . The c o n v e r s e i s c l e a r .  Suppose a%B  B i s commutative and a, B a r e BG-modules. Then  has an BG-module s t r u c t u r e g i v e n by t h e d i a g o n a l a c t i o n  o f G: g (a»b)= (ga)8 ( g b ) , f o r elements g o f G, a o f A, of  B.  If  A  and  b  i s weakly p r o j e c t i v e , then the BG-module A 8 B R  w i t h d i a g o n a l a c t i o n of G  is  also  weakly  projective  [.&,.  P r o p o s i t i o n 8.5].  10.2  Proposition.  Let  B be a commutative r i n g , and G be a  group. (1)  For  < p. d.^H  any  BG-module  M,  p.d. H  < p.d.  R  • c. d.^G.  (2) g l . d . f i < gl.d.BG < g l - d . f i • c.d.^G.  Proof.  It  suffices  to  prove t h a t p.d.^H = 0 i m p l i e s  p . d . ^ f l < n i n the case t h a t c.d.^G = n i s f i n i t e , is  an  BG-module.  The  P r o p o s i t i o n s 9.5 and  rest  B  of  9.6.  Equipped  with  the  diagonal  o f G, t h i s complex becomes an a c y c l i c complex of BG-  modules, w h i l e B8^H term  resolution  l e n g t h n. By B - p r o j e c t i v i t y o f H, the complex  .P.SflM—>B8gM—>0 i s a c y c l i c . action  H  o f the a s s e r t i o n f o l l o w s from  S i n c e c.d.^G = n, t h e r e i s an B G - p r o j e c t i v e .p. o f  where  in  the  i s isomorphic  complex  .P.8^H  t o M as  is  BG-modules.  B-projective  p r o j e c t i v e , hence i s B G - p r o j e c t i v e . T h e r e f o r e  Each  and weakly  .P.« M R  is  an  61 B G - p r o j e c t i v e r e s o l u t i o n of M o f l e n g t h n. Thus p.d.^M <  Section  11. The  In  Finite  Case _And  t h i s s e c t i o n , we  B i s an a r b i t r a r y r i n g and group.  The  Case Of  n.  H-groups  s h a l l i n v e s t i g a t e l . g l . d . B G , where G is a  finite  group  or  an  M-  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)  t o get a s t r o n g e r  result  in  the case t h a t G i s a f i n i t e l y g e n e r a t e d n i l p o t e n t group.  The  following  Proposition  takes  care  of the  finite  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 o f G o f f i n i t e and  index  n,  n i s a u n i t i n B, then l . g l . d . B G = 1.gl.d.BH. (2) I f G i s f i n i t e of o r d e r n, where n i s not a u n i t i n  B, t h e n l . g l . d . B G i s i n f i n i t e .  Proof.  (1)  The  argument o f £ 19, Appendix 2, Lemma 3]  can e a s i l y be m o d i f i e d t o show t h a t e v e r y BG-module which i s BH-projective i s a l s o BG-projective. Propositions  9.5  and  The  r e s u l t f o l l o w s from  9.6.  (2) As n i s not a u n i t i n 8, t h e r e i s a prime f a c t o r of £32,  n  such  that  p i s not a u n i t i n B.  p  By Cauchy's Theorem  Theorem 5 . 2 j , G has a c y c l i c subgroup  H  of  order  p.  62  According  to  Proposition  9.7,  l.gl.d.BH, i s  infinite.  T h e r e f o r e 1.gl.d.BG i s i n f i n i t e .  He now t u r n t o t h e c a s e where G i s an M-group. By 7.1.10],  an  M-group  G  has  a  normal  1=G <G, <-..<G i =G, where n=h (G) , such t h a t G/G e  and  Wf  w  £32, series  is  finite  a l l t h e o t h e r f a c t o r s a r e i n f i n i t e c y c l i c . Such a n o r m a l  s e r i e s w i l l be c a l l e d a s t a n d a r d Suppose  H  is  H-series.  a normal subgroup o f G such t h a t G/H i s  i n f i n i t e . Then o b v i o u s l y  BG i s i s o m o r p h i c , a s r i n g s , t o  the  t  skew  group  defined  ring  by  BH {. x , x * J , _  xa=(a*)x,  where  a*=xa.x~ --. 1  the m u l t i p l i c a t i o n r s Thus  the  following  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 2 3 ] . 11.2.  Proposition.  subgroup o f  G  Let  such  1.gl.d.BG < l . g l . d . B H  11.3.  .Proposition.  s e r i e s 1=G <. .-<G 0  of t h e f i n i t e  ntl  B  be  that  G/H  a  ring  i s infinite,  be a normal  cyclic.  Then  Let  G  be  an  M-group w i t h s t a n d a r d M-  =G. L e t B be a r i n g . Suppose  the  order  group G/G„ i s a u n i t i n B. Then  preceeding  Propositions  H  +1.  l ^ g l . d . B G < 1.g 1.d. B • n = l . g l . d . B  The  and  • h (G).  P r o p o s i t i o n i s a d i r e c t conseguence o f  11.1.(1) and 11.2. As a c o r o l l a r y , we have  following group-theorectic  result.  the  63 11.4.  Proposition.  Let G  s e r i e s 1-G <. . .<G is a  an  M-group w i t h  s t a n d a r d M-  =G such t h a t t h e o r d e r of G/G„ i s m. I f  0  x  be  torsion  element of G of o r d e r k, then every prime  f a c t o r o f k i s a prime f a c t o r o f m.,  P r o o f . Deny t h e c o n c l u s i o n .  Then  there  i s a  cyclic  subgroup fl o f prime o r d e r p such t h a t p i s not a f a c t o r o f m. L e t B be t h e l o c a l i z a t i o n o f the r i n g o f i n t e g e r s a t t h e multiplicative  s e t {1, m, m , . . . j . Then m i s a u n i t i n B,  while p i s not.  By P r o p o s i t i o n  However,  Propositions  a  by  11.3,  1. g l . d . BG  i s finite.  11.1 and 9.5, 1. gl.d.BG i s  i n f i n i t e . This i s a contradiction.  He s h a l l now prove a s t r o n g e r 11.3,  supposing  version  of  Proposition  G i s f i n i t e l y g e n e r a t e d n i l p o t e n t . The key  f a c t t h a t makes t h e 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 : L e t 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  series  l=Z <Z 0  t  homomorphisms  <-..,  ,  £Z *i /Z„,Z j ,  Then  (  n  o f the p a i r o f a b e l i a n  the  of  groups Zn+//Z„ and Z,,  n=1,2,-.., s e p a r a t e s p o i n t s . I n o t h e r words, f o r i n Znti /Z„,  set  any Z„x#1  t h e r e i s f i n .[Z * i /Z ,Z, ] such t h a t f ( Z x ) f 1 i n M  n  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 t h e o r d e r o f every t o r s i o n element o f 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) F o r k=1,.,.,n, i f Zk/Z«., has a t o r s i o n element o f prime  order  p, t h e n  Z, h a s a t o r s i o n element o f o r d e r p.  C o n s e q u e n t l y , f o r any r i n g 8, Z , i s  fi-torsion-free  implies  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 , g e n e r a t e d 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 finitely  generated,  then the s e t T o f  t o r s i o n e l e m e n t s o f 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 o f  G. (4) . I f G i s f i n i t e l y g e n e r a t e d , t h e n G h a s an H - s e r i e s 1=G,<Gi<..,<G=G  such  W  f a c t o r s are i n f i n i t e  that  G  (  i s finite  and the o t h e r  cyclic.  P r o o f . (1) f o l l o w s  from  that  [Zu/Z*., , Z i ] s e p a r a t e s  points. (2) finitely free  by  Since  G  h a s maximal  c o n d i t i o n , each Zk/Z ., i s k  generated a b e l i a n * These f a c t o r s a r e a l s o  torsion-  ( 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 w i t h o n l y i n f i n i t e c y c l i c  factors.  (3) The s e t o f t o r s i o n e l e m e n t s o f 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 current  assumptions,  T  133, 6.4.13 J.  Under t h e  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 e t Gi=T- Then G/G, i s t o r s i o n - f r e e . The c o n c l 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 o f [14,  Theorem  5,  P-149J) L e t G be a f i n i t e l y g e n e r a t e d n i l p o t e n t group, and B be a r i n g . (1)  If  G  i s not  B-torsion-free,  then  c.d. G R  is  infinite. (2) I f G i s B - t o r s i o n - f r e e , t h e n c.d.RG = h (G).  11.8. P r o p o s i t i o n . L e t G be a f i n i t e l y  generated  nilpotent  group, and fl be a r i n g . (1)  l . g l . d . B G i s f i n i t e i f and o n l y 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. G. R  Proof. has  (2) S i n c e G i s f i n i t e l y  finite  Hirsh  1=Ga<G/<...<Gj 5) , =G t(  infinite  +  number with  G i  generated n i l p o t e n t , i t  h(G)  and  an  M-series  f i n i t e and t h e o t h e r  c y c l i c . By P r o p o s i t i o n s 11.1  and  11.2,  factors  l.gl.d.BG  < l . g l . d . B • h(G) •= l . g l . d . B + c.d.^G. A l s o c.d.RG < l . g l . d . B G by  definition., (1)  If  G  i s B-torsion-f ree,  then  (2)  shows  l.glwd.BG i s f i n i t e . I f G i s n o t B - t o r s i o n - f r e e , then G a  finite  that has  c y c l i c subgroup H of o r d e r n o t a u n i t i n B. S i n c e  l . g l . d . B H i s i n f i n i t e , so i s l . g l . d . B G .  66  S e c t i o n 12. O s o f s k y ' 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  following  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 .  12.1.  £Qsofsky's Theorem, 2 9 , Theorem 2 . 4 4 and  Proposition.  2 . 3 8 J L e t S be a r i n g . L e t { A;3j  Proposition system  be  a  direct  o f S-modules indexed by a s e t I o f c a r d i n a l i t y <  .  Then p . d . ^ l i n ^ { A j } < sup {p.d- A- ) *af 1* T  Let  {G;}  be  x  - l i a , { G, j . r  5  Let  a H £  £  x  to  get  an  system  x  of a  direct  groups  and  G  system  of  BG-modules  BG-module. But t h e r e i s a n o t h e r type o f  d i r e c t l i m i t which we a r e i n t e r e s t e d direct  of  be a r i n g . Of c o u r s e one c a n take t h e  d i r e c t l i m i t lim^ f A j ( A }  direct  r  i n : Let  { Mjjj.  be  a  system o f B-modules o v e r t h e d i r e c t system o f groups  £ G ^ j j . ( T h i s means t h a t M i i s an BG;.-module f o r each  i  in  I , t h a t { M-}j i s a d i r e c t system of B-modules, and t h a t t h e BG;-module  structures  d i r e c t systems  { G -j  = lim., f -H(lj  has  t  are and  z  a  c o m p a t i b l e w i t h t h e maps i n t h e £ M j .) t  Then  x  natural  the  BG-module  d i s t i n g u i s h t h i s BG-module from t h e u s u a l  B-module  structure.  direct  limit  M To of  BG-modules, we denote i t by L i m £ M i 3 . >  Let  £ Mjj  and  x  { G-J  p a r a g r a p h . Then { BG B^.H^jj modules,  where  G  x  x  be  is a  = link, {G^.Jj...  as  i n the  direct  preceeding  system  of  A r e s u l t o f Gruenberg  BGsays  t h a t l i m £ EG ® ^ M ; J  Z  = Lim^ ( M '3 t  £ 1, V  as BG-modules  x  Lemma 2 j .  12.2. P r o p o s i t i o n . are = l i %  inclusions,  L e t { <-i} be a d i r e c t s y s t e m , whose maps z  of  subgroups  of  a  group G, such t h a t G  £ G j } - Suppose t h e c a r d i n a l i t y o f I i s <  Then:  r  (1)  For  < sup [p. d. (2)  any  BG-module  p.d.^M  M} • n • • 1. r  l . g l . d . B G < sup { l . g l . d . BGi }  Proof.  H,  (1) S i n c e  G  + n + 1.  J  i s a subgroup o f 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 modules  over  result,  M  the d i r e c t = Lirn^ f H J  r  system  £ G j j j . ,. By  = lirn^ (RG  flat  ( r i g h t ) BG^-module. T h e r e f o r e  Gruenberg's  j - , By r  Theorem, p.d.^M < sup {p.d.^RG ®^,a j  r  o f B-  Osofsky*s  * n • 1. But BG i s a  p. d.^BG  < p.d.^M  f o r each i . The p r o o f i s completed. (2) i s an immediate conseguence o f ( 1 ) .  12.3. P r o p o s i t i o n .  Let G  be  a  locally  finite  group o f  c a r d i n a l i t y < $n . L e t B be a r i n g * (1) l . g l . d . B G i s f i n i t e i f and  only  i f l.gl.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 o n - f r e e , then l . g l . d . B  l.gl-d.BG  < l . g l . d . B + n • 1.  Proo. G i s t h e d i r e c t l i m i t o f  the d i r e c t  system  of  68  finitely  generated  subgroups.  The o r d e r o f e v e r y f i n i t e l y  generated subgroup i s a u n i t i n B i f and o n l y torsion-free. and  The  if  G  is  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  R-  12.2(2)  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 e v e r y f i n i t e l y generated  subgroup  of G has a s t a n d a r d M - s e r i e s , such t h a t  of  the  order  its  f i n i t e f a c t o r i s a u n i t i n R, t h e n : l . g l . d . f i < l.gl.d.BG < l - g l . d . B  P r o o f . , Under  the  • h (G) .• n > 1.  assumptions,  f i n i t e l y generated subgroup H  of  G,  we  have,  l.gl.d.BU  + h ( G ) . O s o f s k y ^ s Theorem i m p l i e s t h e d e s i r e d  Again,  the  preceeding  result  when G i s l o c a l l y n i l p o t e n t . We properties  first  for  every  < 1 . g l . d.B  inequality.  has a s t r o n g e r v e r s i o n make  explicit  of l o c a l l y n i l p o t e n t groups which w i l l be  some useful  later,  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 t h e s e t of i t s t o r s i o n elements* (1)  T  is  a  locally  c h a r a c t e r i s t i c subgroup  of  finite  locally  nilpotent  G.,  (2) h(G) = h (G/T). (3)  If  G  is  l o c a l l y n i l p o t e n t o f c l a s s l e s s t h a n or  e q u a l t o a f i x e d number k, then G i s n i l p o t e n t o f c l a s s l e s s  69  than o r e q u a l t o k.  P r o o f . (1) G i s a c t u a l l y n i l p o t e n t by d e f i n i t i o n .  locally  finitely  generated  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 t h a t 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 g e n e r a t e d subgroup a o f G, t h e r e  c o r r e s p o n d s a f i n i t e l y g e n e r a t e d subgroup equal  Hirsh  number  converse a l s o  holds*.  (h (H)  flT/T  = h (H/H D T)  Therefore  h(G)  of  G/T  of  = h (HT/TJ) .  The  = h (G/T) ; by  taking  suprema. (3)  Induce  on k. Suppose k = 1. G i s l o c a l l y  hence a b e l i a n . Suppose t h e a s s e r t i o n groups  locally  nilpotent  of  has  class  been  abelian,  proved  for  < k, and G i s a group  l o c a l l y n i l p o t e n t o f 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 o f c l a s s < k. T h e r e f o r e G has a f i n i t e l o w e r c e n t r a l s e r i e s o f l e n g t h < k.  12*6. 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 cardinality  <  .  following assertions  group  L e t T be t h e t o r s i o n subgroup  with  o f G. The  are equivalent:  (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 nilpotent,  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  finite. (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  torsion-free. (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 t h a t h(G) < c.d.^G, and t h a t ,  i f G i s B - t o r s i o n - f r e e , c. d.^ G < h (G) • n • 1. To see let  { Gj,jj  be  t h e d i r e c t system o f the f i n i t e l y  subgroups o f G. Then h(G) = sup £ h(G;) } < c.d.gG. 12.2(1)  Suppose  G  - sup { h(G;) j  x  c.d. G  < sup {  ft  generated  < sup { c.d.^Gj J  z  i s B-torsion-free.  11.7(2),  and  this,  By c,&.  Propositions Gi}  R  x  z  • n • 1 = h (G) • n «• 1. Of c o u r s e ,  • n  +1  when  G  i s n o t 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 . , It  f o l l o w s from t h i s o b s e r v a t i o n  i f and o n l y i f h(G) Therefore  i s finite  and  t h a t c.d. G i s f i n i t e R  G  is  B-torsion-free.  (2) i s e q u i v a l e n t t o ( 3 ) , and (4) i s e q u i v a l e n t t o  (5). By  Proposition  is finite* class  12-5(2), h (G) i s f i n i t e i m p l i e s h (G/T)  Being t o r s i o n — f r e e , G/T i s l o c a l l y  nilpotent  of  l e s s than o r e q u a l t o 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 p r o v i d e d  h(G) i s f i n i t e .  I t remains t o show t h a t l . g l . d . B G i s f i n i t e i f and o n l y i f l . g l . d . B and c.d.^G a r e b o t h f i n i t e . O b v i o u s l y , l . g l . d . B G i s f i n i t e i m p l i e s b o t h l . g l . d . B and c. d.^G are reverse  12.7.  The  i m p l i c a t i o n f o l l o w s from t h e f o l l o w i n g i n e q u a l i t y .  Proposition.  Let G  be  a locally  c a r d i n a l i t y < tf . Suppose a l s o t h a t n  Then  finite.  h(G)  < c.d.^G  < l . g l . d. BG  G  n i l p o t e n t group o f is  < l.gl.d.B  B-torsion-free., * h (G)  + n + 1  71  < l . g l . d . B + c.d.^G + n * 1.  P r o o f . He have shown t h a t h<G) < c.d.^G. L e t { G-.} t  the By  r  be  d i r e c t system o f t h e f i n i t e l y g e n e r a t e d subgroups o f G. the  results  we  < sup { l.gl.d.BG^ j  r  have  proved  • n • 1  * n + 1 = l . g l . d . B + h(G)  so  far,  l.gl.d.BG  < sup { l . g l . d.B+h (G;) }j  n +1. The p r o o f i s completed.  72 REFERENCES  1  Anderson, F.W. and F u l l e r , K . f i . , B i n g s and Categories of n o d u l e s , Grad. 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