UBC Theses and Dissertations

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UBC Theses and Dissertations

Torsion and localization Vilciauskas, Algis Richard 1972

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TORSION AND LOCALIZATION by A l g i s Richard V i l c i a u s k a s B . S c , Loyola of Montreal, 1970 •A THESI'S SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE •f -v-i -J- s-i T*\ .-N t~, • v- 4- w i y-» +» \ U-LX. C U V - i-» C - C L i - t, t u v - i i i_ v of MATHEMATICS We accept t h i s t h e s i s as conforming to the req u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA J u l y , 1972 In p re sen t i ng t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and s tudy . 1 f u r t h e r agree tha t pe rmiss ion fo r e x t e n s i v e copying o f t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . It i s understood that copying or p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l ga in s h a l l not be a l lowed wi thout my w r i t t e n p e r m i s s i o n . Department o f MATHEMATICS The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8, Canada Date July 25, 1972 ABSTRACT The purpose of t h i s t h e s i s i s to develop the machinery of -noncommutative l o c a l i z a t i o n as i t i s being used to date, along w i t h some fundamental r e s u l t s and examples. We are not concerned w i t h a search f o r a "true t o r s i o n theory" f o r R-modules, but r a t h e r w i t h a u n i f i c a t i o n of previous g e n e r a l i s a t i o n s i n a more n a t u r a l c a t e g o r i c a l s e t t i n g . In s e c t i o n 1, the g e n e r a l i s a t i o n of t o r s i o n f o r a r i n g R manifests i t s e l f as a k e r n e l f u n c t o r which i s a l e f t exact subfunctor of the i d e n t i t y f u n c t o r on the category of R-modules. I f a k e r n e l f u n c t o r a a l s o has the property a(M/o(M)) = 0 f o r any R-module M- , we- say-that' o" i s * idempotrent--;--We--treat- fche>-C-abr-ial.-corr-espondence-•• which e s t a b l i s h e s a c a n o n i c a l b i s e c t i o n between k e r n e l f u n c t o r s , f i l t e r s of l e f t i d e a l s i n R , and c l a s s e s of R-modules c l o s e d under submodules, extensions, homomorphic images, and a r b i t r a r y d i r e c t sums. This r e s u l t , which allow s us to view t o r s i o n i n s e v e r a l e q u i v a l e n t ways, i s fundamental to the r e s t of• the t h e s i s . S e c t i o n 2 presents some p o s i t i v e and negative observations on when a k e r n e l f u n c t o r i s idempotent. In s e c t i o n 3 we begin by g e n e r a l i s i n g the concept of i n j e c t i v e module by d e f i n i n g o - i n j e c t i v i t y r e l a t i v e to an idempotent k e r n e l f u n c t o r a . This y i e l d s a f u l l c o r e f l e c t i v e subcategory of the category of R-modules. The l o c a l i z a t i o n f u n c t o r r e l a t i v e to a i s then constructed as the composite of the c o r e f l e c t o r w i t h the embedding of the subcategory. I n s e c t i o n 4 we d i s c u s s the important "property T" which allows us to express the l o c a l i z a t i o n of an R-module as the module tensored w i t h the l o c a l i z e d r i n g , j u s t as i n the c l a s s i c a l commutative case of l o c a l i z i n g at a prime i d e a l . F i n a l l y i n s e c t i o n 5 we see that every idempotent k e r n e l f u n c t o r can be represented by a f i n i t e l y cogenerating i n j e c t i v e R-module V and the r e l a t i v e l o c a l i z a t i o n of R -by the double c e n t r a l i z e r of V . I n d i c a t i o n s are th a t the g e n e r a l i s e d concept of t o r s i o n w i t h i t s r e l a t i v e l o c a l i z a t i o n w i l l prove i t s e l f i n c r e a s i n g l y v a l u a b l e i n the f u r t h e r study of r i n g s and modules. V. TABLE OF CONTENTS INTRODUCTION 1 Se c t i o n 1. TORSION THEORIES 4 Se c t i o n 2. KERNEL FUNCTORS ' 15 Sec t i o n 3. LOCALIZATION FUNCTORS 22 Sec t i o n 4. PROPERTY T 32 Se c t i o n 5. REPRESENTATION OF IDEMPOTENT KERNEL FUNCTORS AND THEIR RELATIVE LOCALIZATIONS 42 BIBLIOGRAPHY 54 v i . ACKNOWLEDGEMENTS • The author wishes to express h i s thanks to h i s s u p e r v i s o r Dr. -S. Page f o r suggesting the t o p i c of t h i s t h e s i s and f o r h i s h e l p f u l advice and encouragement during i t s p r e p a r a t i o n . The author a l s o wishes to thank Dr. D. C. Murdoch f o r h i s c a r e f u l reading of t h i s t h e s i s . The most generous f i n a n c i a l support of the N a t i o n a l Research C o u n c i l of Canada i s g r a t e f u l l y acknowledged. 1. INTRODUCTION To every a b e l i a n group G we can a s s i g n a t o r s i o n subgroup T(G) c o n s i s t i n g of a l l the elements of G w i t h f i n i t e order. I f G' <S G i s a subgroup, i t i s c l e a r that T(G') = G' D T(G) . Furthermore, any group homomorphism G > H n e c e s s a r i l y maps T(G) i n t o T(H) . Thus we may regard T as a l e f t exact subfuhctor of the i d e n t i t y functor on the category of a b e l i a n groups. This f o r m u l a t i o n of the usu a l t o r s i o n theory i n a b e l i a n groups lends i t s e l f e a s i l y to a g e n e r a l i s e d concept of t o r s i o n f o r other "nice", a b e l i a n c a t e g o r i e s . In t h i s t h e s i s we s h a l l be concerned only w i t h the g e n e r a l i s a t i o n to ca t e g o r i e s of modules over an a r b i t r a r y r i n g w i t h u n i t y . A l e f t exact subfunctor"of" the'- 4identity" on- M« ±s-• GaM-ed^ a-k-e-r-nel--f«uno-t-o.E*.-:-A k e r n e l f u n c t o r a f o r which a(M/a(M)) = 0 f o r every R-module M w i l l be c a l l e d idempotent. Thus the usual t o r s i o n theory i n a b e l i a n groups i s a prototype f o r our idempotent k e r n e l f u n c t o r s defined on more general module c a t e g o r i e s . However, we are not concerned w i t h a search f o r a "true t o r s i o n theory" f o r R-modules, but r a t h e r w i t h a u n i f i c a t i o n of previous g e n e r a l i s a t i o n s i n a more n a t u r a l c a t e g o r i c a l s e t t i n g . In s e c t i o n 1 we t r e a t the G a b r i e l correspondence which e s t a b l i s h e s a c a n o n i c a l b i j e c t i o n between k e r n e l f u n c t o r s , f i l t e r s of l e f t i d e a l s i n R , and c l a s s e s of R-modules closed under submodules, exte n s i o n s , homomorphic images, and a r b i t r a r y d i r e c t s u m s I n the case of idempotent k e r n e l f u n c t o r s , t h i s b i j e c t i o n r e s t r i c t s to a s i m i l a r correspondence which enables us to view the g e n e r a l i s e d concept of 2. t o r s i o n i n s e v e r a l e q u i v a l e n t ways. The G a b r i e l correspondence i s fundamental to the r e s t of t h i s t h e s i s . S e c t i o n 2 presents some p o s i t i v e and negative observations on when a k e r n e l f u n c t o r i s idempotent. This i s done mainly by i n v e s t i g a t i n g the a s s o c i a t e d f i l t e r of l e f t i d e a l s . In s e c t i o n 3 we begin by g e n e r a l i s i n g the concept of i n f e c t i v e module. This i s done r e l a t i v e to any idempotent k e r n e l f u n c t o r a by remodeling the i n f e c t i v e t e s t lemma i n the sense t h a t a module A i s c a l l e d a-~infective i f the ex t e n s i o n property enunciated i n the i n f e c t i v e t e s t lemma holds f o r at l e a s t the l e f t i d e a l s i n the f i l t e r a s s o c i a t e d w i t h a . I f the extensions are unique, we say that the module A i s f a i t h f u l l y a - i n j e c t i v e . Now the l o c a l i z a t i o n f u n c t o r r e l a t i v e to <T as's±gn's~"to—ea-ch- R-modulre'-its' f^±^1^117*,o--iUj'e"c"t'ive---hu±'r;'' • In" order" t r c o n s t r u c t t h i s f u n c t o r e x p l i c i t l y , we consider the f u l l subcategory of c o n s i s t i n g of the f a i t h f u l l y a - i n j e c t i v e R-modules. This subcategory i s c o r e f l e c t i v e w i t h exact c o r e f l e c t o r , and the l o c a l i z a t i o n f u n c t o r i s the composite of the c o r e f l e c t o r w i t h the embedding of the subcategory. The l o c a l i z a t i o n of the r i n g R i s again a r i n g , but now need no longer be a l o c a l r i n g i n the sense of having a unique maximal i d e a l . E q u ivalent formulations of t h i s l o c a l i z a t i o n process are a l s o mentioned. In s e c t i o n 4 we d i s c u s s the important "property T" which allows us to express the l o c a l i z a t i o n of any R-module as the module tensored w i t h the l o c a l i z a t i o n of R , j u s t as i n the c l a s s i c a l commutative case of l o c a l i z i n g at a prime i d e a l . In s e c t i o n 5 x\re see that every idempotent k e r n e l f u n c t o r a can be represented by a f i n i t e l y cogenerating i n j e c t i v e R-module V ( where f i n i t e l y cogenerating i s the dual of f i n i t e l y generated ) i n the sense that a(M) i s the i n t e r s e c t i o n of the k e r n e l s of a l l R-homomorphisms of M i n t o V f o r any M i n M , Furthermore the K— l o c a l i z a t i o n of R r e l a t i v e to a i s the double c e n t r a l i z e r of V . A l l r i n g s have u n i t y 1 and a l l r i n g morphisias are u n i t a l . The category of l e f t u n i t a r y modules over a r i n g R i s denbted by ^M ( M^ f o r r i g h t R-modules ). Morphisms i n are c a l l e d R-maps. Module always means l e f t unless s t a t e d otherwise. For any R-module M w i t h submodules M1,M" we use the n o t a t i o n (M':M") f o r the l e f t i d e a l { r e R | rM" Q M'} . Thus i n p a r t i c u l a r (0:m) i s the a n n i h i l a t o r of m e M , and (0:M) the a n n i h i l a t o r of M . I(M) denotes the- i n j ee-feiv.e.-hu«M*- of-- M»-.- T.he--hom^f.unGtor* in,--at Ga-fcegory-- •  _G-.. is.f denoted,, by £,(.]_>]_)• The s i t u a t i o n of a fu n c t o r F being l e f t a d j o i n t to a f u n c t o r G i s denoted by F — I G . Proof of r e s u l t s are given e i t h e r when they could not be found i n the l i t e r a t u r e , or an a l t e r n a t e proof i s o f f e r e d . Otherwise a refe r e n c e i s given. The symbol \ i n d i c a t e s the end of a proof. E f f o r t has been made to i n d i c a t e as much as p o s s i b l e the source of terminology used i n t h i s t h e s i s , and to mention other terminology used elsewhere. The b a s i c references throughout are [7,9,16,17,22,33]. 1. TORSION THEORIES Let be the category of l e f t modules ( w r i t t e n M f o r short ) over a r i n g R w i t h u n i t y 1. A subfunctor of the i d e n t i t y on M i s a c o v a r i a n t endofunctor a : M — y M such that a(M) Q M i s a submodule f o r every M e M and a ( f ) : a(N,) > a(M) i s the r e s t r i c t i o n f o r any f : N > M i n M . (1.1) D e f i n i t i o n ; A subfunctor a of the i d e n t i t y on M i s c a l l e d a k e r n e l f u n c t o r [9] i f a i s l e f t exact. E q u i v a l e n t l y , a i s a k e r n e l f u n c t o r i f a i s a subfunctor of the i d e n t i t y on M such that a(N) = N C) a(M) f o r any submodule N of M e M . An idempotent k e r n e l  f.unc.t.o.r„.[.9.],.. is,,.a..kernel, functor, a_. s a t i s f y i n g , a(M/,a,(M).)„. = 0, .. We denote the f a c t that a : M -—> M i s a k e r n e l f u n c t o r ( idempotent k e r n e l f u n c t o r ) by a e KF(R) ( a e IKF(R) ) . We have the c l a s s i n c l u s i o n IKF(R) *C KF(R) which w i l l be shown to be set i n c l u s i o n . For a e KF(R) and any M e M we c a l l o(M) the a - t o r s i o n submodule of M. Observe that i f a e KF(R) then a 2 = a i e . a(a(M)) = a(M) A a(M) = a(M) . Various endo-functors* of; M having'an- assortment of- d i f f e r e n t names appear i n the l i t e r a t u r e : a subfunctor of the i d e n t i t y on M i s c a l l e d a p r e r a d i c a l i n [5,21]; a k e r n e l f u n c t o r i s c a l l e d a concordant i n [33]; a subfunctor a of the i d e n t i t y on M such that a(M/0(M)) = 0 f o r every M z M i s c a l l e d a r a d i c a l i n [5,17] where a l s o an idempotent k e r n e l f u n c t o r i s c a l l e d a t o r s i o n r a d i c a l . (1.2) D e f i n i t i o n : A f i l t e r of l e f t i d e a l s [9,33] i n a r i n g R i s a s e t of l e f t i d e a l s F. s a t i s f y i n g the f o l l o w i n g c o n d i t i o n s : i ) i f U e F_ and I i s a l e f t i d e a l c o n t a i n i n g U then I e F_ i i ) i f U,V e F_ then U O v e F i i i ) i f r e R and U e F then (U:r) e F . These f i l t e r s were considered by G a b r i e l [2,7] where such an o b j e c t was c a l l e d un ensemble d'ideaux a gauche t o p o l o g i s a n t . I f i n a d d i t i o n a f i l t e r s a t i s f i e s : i v ) i f I i s a l e f t i d e a l f o r which there e x i s t s some U e F_ w i t h (I:u) e F_ f o r every u e U then I e F_ then the f i l t e r F_ i s c a l l e d s t r o n g l y complete [33]. G a b r i e l [2,7] c a l l s such an o b j e c t un ensemble d'ideaux ( a gauche ) t o p o l o g i s a n t ~ et id'empotent"'. (1.3) D e f i n i t i o n : A Serre c l a s s [33] i n RM i s a non-empty subclass of M such that i f 0 > A' y A > A" 0 i s exact then A',A" e S_ i f and only i f A e _S . E q u i v a l e n t l y , a Serre c l a s s S i s a non-empty subclass of M c l o s e d under submodules, — K— homomorphic images, and extensions. An a d d i t i v e c l a s s [33] of R-modules i s a non-empty subclass of M c l o s e d under submodules, homomorphic images, and f i n i t e d i r e c t sums. We say that a c l a s s of R-modules i s s t r o n g l y complete [33] i f i t i s c l o s e d under a r b i t r a r y d i r e c t sums. These Serre c l a s s e s , s t r o n g l y complete a d d i t i v e c l a s s e s , and s t r o n g l y complete Serre c l a s s e s are e x a c t l y l e s sous-categories e'paisse, l e s sous-categories fermees, and l e s sous-categories l o c a l i s a n t e r e s p e c t i v e l y , considered i n [ 7 ] . 6. In the f o l l o w i n g paragraphs we want to give e x p l i c i t l y the G a b r i e l correspondence which was f i r s t announced ( p a r t i a l l y ) i n [7, Chap.5] Let us denote the set of f i l t e r s on R by FI L ( R ) . For any a e KF(R) , put F a = { I £ l e f t i d e a l s of R | a ( R / l ) = R / l } Then the mapping a >—> F^ d e f i n e s a c a n o n i c a l b i j e c t i o n between KF(R) and FIL(R) by [9,Thms. 2.1,2.2] w i t h the i n v e r s e mapping f o r any F_ e FIL(R) given by . F_ i >• T where x (M) = { m e M | (0:m) e F_ } f o r any M e M having the property that F_ = F^ and a (M) = { m e M | (0:m) £ F^} From t h i s i t f o l l o w s immediately that KF(R) forms a s e t , as observed i n - [-9Y1"3-]T and- that* • lKF-(4V)-# KF-(R>-- i s - s e t - inclusdronv (1.4) Lemma: o e KF(R) i s Idempotent i f and only i f F i s a strongly complete f i l t e r . Proof; This I s - e x a c t l y the content of [9,Thm 2.5]. Next l e t us denote the c l a s s of a l l s t r o n g l y complete a d d i t i v e c l a s s e s i n „M by CAD(R) . For any F £ FIL(R) put S„ = { M e M I (0:m) E F f o r every m e M }" Then the mapping F_ i—> d e f i n e s a c a n o n i c a l b i j e c t i o n between FIL(R) and CAD(R) by [33,Thm 1.10], w i t h the i n v e r s e mapping f o r any _S e CAD(R) given'by 1' y = ^ 1 e l e f t i d e a l s of R | R / l e S } having the property that S_ = j and F_ = E^ g J From [33,Lemma 1.18] we have (1.5) Lemma: F_ e FIL(R) is strongly complete i f and only If i s a strongly complete Se.rre class. The correspondences given above induce a c a n o n i c a l b i s e c t i o n between KF(R) and CAD(R) which i s e a s i l y computed to be the mapping a i — • S = { M e M | a (M) = M } w i t h i n v e r s e given by S i > T where T (M) = \ { ImOjO | Tp e M(S,M) , S e S_ } The members of S are c a l l e d a - t o r s i o n modules. Hence a(M) i s the -a • ;  l a r g e s t ( n e c e s s a r i l y unique ) a - t o r s i o n submodule of M. Combining Lemmas (1.4,1.5) we have (1.6) Lemma: a e KF(R) is idempotent if and only if is a strongly complete Serre class. C o l l e c t i n g these r e s u l t s , we s t a t e (1.7) Theorem: ( Ga b r i e l , correspondence f o r M ) : K.— i ) . There is a canonical bijection between kernel functors, filters of left ideals, and strongly complete additive classes. , i i ) Restriction of the bijection in i ) yields a canonical bijection between idempotent kernel functors, strongly complete filters of left ideals, and strongly.complete Serre classes. There i s another object a s s o c i a t e d w i t h every a e IKF(R) th a t turns out to be important l a t e r when c o n s i d e r i n g l o c a l i z a t i o n s , namely (1.8) V = { M - e M | a(M) = 0 } . . The c l a s s of R-modules i s closed under isomorphic images, submodule d i r e c t products, and i n j e c t i v e h u l l s by [17,Prop 0.3], and i t s members are c a l l e d a - t o r s i o n - f r e e modules. Moreover, by the same P r o p o s i t i o n we have (1.9) S g = -{ M e M.| M(M,A) = 0 f o r every A e Vff } . This c o i n c i d e s w i t h the c l a s s i c a l n o t i o n i n a b e l i a n groups that we cannot map a t o r s i o n group i n t o a t o r s i o n - f r e e group i n a n o n - t r i v i a l way. (1.10) P r o p o s i t i o n : For any a e IKF(R) ' Ya is a full coreflective subcategory of M with coreflector F : M >- whose object function is given by F(M) = M/a(M) . Furthermore, i s an abelian category, and the coreflector F is exacts Proof: Let K : V > M be the embedding funct o r of V considered - — 6 -a as a f u l l subcategory. I t should f i r s t be remarked that monomorphisms i n V and i n M c o i n c i d e , -a — I f f : M >• N i n M , then the diagram 0 — > a(M) — y M -—> M/a(M) = F(M) — y 0 ( i . i i ) a ( f ) f : : 0 — y a(N) — y N — > N/a(N) = F(N) — y 0 has exact rows. We d e f i n e F ( f ) as the unique f a c t o r i z a t i o n of f. over the cokernels making (1.11) commute. This makes . F i n t o a f u n c t o r . For any M e M , C e , and TT^ : M > M/a(M) c a n o n i c a l i r ^ : V (FM,C) y M(M,KC) i s c l e a r l y a n a t u r a l isomorphism by (1.9). Thus ir : M y F(M) i s a c o r e f l e c t i o n f o r M i n [22,p.128]. Suppose f i n (1.11) i s a mono and that F ( f ) ( m + a(M)) = 0 f o r m e M . As the diagram commutes, we have f(m) e a(N) . So there i s a U e F q such that Uf(m) = 0 . This i m p l i e s f(Um) = 0 , and f mono i m p l i e s Um = 0 . Hence m e a(M) and 9. so F ( f ) i s a mono. Hence by [22,Prop 5.3,p.130] i s an a b e l i a n category and by [22,Prop 12.1,p.67] F i s exact. 1 The G a b r i e l correspondence s t a t e d i n (1.7) allows us to view the g e n e r a l i s e d concept of t o r s i o n i n s e v e r a l equivalent ways. Given a e IKF(R) we s h a l l c a l l the t r i p l e of a s s o c i a t e d o b j e c t s ( F q , S , V ) a t o r s i o n theory. These ideas appear much more concrete a f t e r i n v e s t i g a t i n g a few s p e c i a l cases. (1.12) Example: Let R be any r i n g , S C R a m u l t i p l i c a t i v e l y c l o s e d system ( i e . S]_ S2 £ $ i f s]_>s2 e s )• L e t <± be t u e s e t °f l e f t i d e a l s I of R such t h a t f o r any r e R there e x i s t s s e S v/ith sr £ I . E q u i v a l e n t l y , G_ = {'" I e l e f t I d e a l s of" R J (I':r) H S i 0" f o r any r e R } . The c o n d i t i o n s f o r a s t r o n g l y complete f i l t e r are e a s i l y seen to be s a t i s f i e d by G_ . For i n s t a n c e to see that c o n d i t i o n i v ) holds l e t J be a l e f t i d e a l such that (J:u) £ G_ f o r every u £ U w i t h U £ G_ . For any r e R , we have that s^r £ U f o r some s^ £ S . Then (J:s2'0 £ £ so t h a t si s2 r e ^ ^ o r s o r a e s^ e S . As S 2 S 2 e ^ we have J e G^  . Let a be the idempotent k e r n e l f u n c t o r corresponding to G_. An R-module H i s a - t o r s i o n i f and only i f f o r every h e H , sh = 0 f o r some s e S , and a(M) i s the l a r g e s t a - t o r s i o n submodule of M f o r any M e M . I f 0 E S e v e r y t h i n g i s t o r s i o n , but assuming 1 £ S changes nothing. Even though every I e G_ meets S, a(M) i s not the subset of M c o n s i s t i n g of elements k i l l e d by some element of S. This set i s 10. not even a submodule i n g e n e r a l . The reason i s that not every s e S need be contained i n an i d e a l belonging to G_. For t h i s to happen we need a "common l e f t m u l t i p l e p r o p e r t y " : (1.13) \/ s e -S \ / r e R 3 t e S 3r' e R . > t r = r ' s ' . This i m p l i e s that every l e f t i d e a l that meets S i s contained i n G_. The c o n d i t i o n (1.13) i s t r i v i a l l y s a t i s f i e d i f S i s c e n t r a l i n R, which i s c e r t a i n l y the case i f R i s commutative. In case S c o n s i s t s of a l l non-zero d i v i s o r s of R, (1.13) i s e x a c t l y the c l a s s i c a l l e f t Ore c o n d i t i o n [16,p.109]. For an e n t i r e r i n g R, the set of a l l non-zero l e f t i d e a l s forms a s t r o n g l y complete f i l t e r i f and only i f R i s a l e f t Ore r i n g . Now i f S C R i s a m u l t i p l i c a t i v e l y c l o s e d system s a t i s f y i n g (1.13) i t i s c l e a r that a(M)" = C m e M' |' sm = 0 f o r some s e' S~ }" f o r any MTe H d e f i n e s an idempotent k e r n e l f u n c t o r w i t h t o r s i o n theory F = { I e l e f t i d e a l s of R I I H S 1 0 } -a 1 S = { H" e M | (0 :h) (~\ S ± 0 f o r any h e . H } V = { C e M | (0:c) (~\ S = 0 f o r a l l 0 + c e C } . For the remainder of t h i s s e c t i o n we s h a l l i n v e s t i g a t e two more s p e c i a l t o r s i o n t h e o r i e s . For any R-module M d e f i n e the l e f t s i n g u l a r submodule Z X(M) as : (1.14) Z (M) = { m e M | (0:m) i s e s s e n t i a l l e f t i d e a l i n R } In case M = R , Z^(R) i s a 2-sided i d e a l [16,p.106 f f ] . The e s s e n t i a l l e f t i d e a l s of a r i n g R ( which p l a y an important r o l e i n the G o l d i e Theory - see f o r example [11] ) form a f i l t e r which i s not i n general s t r o n g l y complete. This i s because Z^(R/Z^R) need not be zero. But we can e i t h e r s h r i n k or enlarge t h i s set of l e f t i d e a l s ( apart from the obious extreems ) so t h a t we do o b t a i n a s t r o n g l y complete f i l t e r . This was b a s i c a l l y the approach of Dlab [ 6 ] . We s t a r t by e n l a r g i n g the set of e s s e n t i a l s . (1.15) Lemma: For a left ideal L of R, the following conditions are equivalent: i ) there exists an essential left ideal E in R such that (L:x) i s essential in R for every x e E i i ) for every r e R with r i L there exists s e R such that (L:sr) i s proper ( ie. R ) essential in R . Proof: I ) => i i ) " IF' r "^'L , Rr'*f*(T so" Rr" -E't 0* implTes 0 ^ s r e E f o r some s e R , and we can p i c k i t so that sr i L ( s i n c e otherwise (E:r) = (L:r) , making (L:r) e s s e n t i a l a l r e a d y ). Then (L:sr) i s proper e s s e n t i a l . i i ) => i ) Let S = { s e R | (L:s) i s e s s e n t i a l }. S ± 0 by t a k i n g r = 1 i n c o n d i t i o n i i ) . Let E be the l e f t i d e a l generated by L and S. I f 1 ^ 0 i s any l e f t i d e a l i n R such that I f ) L = 0 , then f o r 0 ^ a e I there e x i s t s b s R such th a t (L:ba) i s proper e s s e n t i a l . Then ba ^ 0 , ba e I and ba e S C E . Hence E i s e s s e n t i a l . I f (L:x) and (L:y) are e s s e n t i a l , then (L:x + y) i s e s s e n t i a l as i t contains (L:x) C\ (L:y) . For any r e R and s e S (L:rs) = ( ( L : s ) : r ) i s e s s e n t i a l . Hence (L:x) i s e s s e n t i a l f o r every x e E . J C l e a r l y any e s s e n t i a l s a t i s f i e s the c o n d i t i o n s of the Lemma. 12. I t was shown by A l l n ( found i n [31] ) that c o n d i t i o n i ) of the Lemma c h a r a c t e r i s e s the l e f t i d e a l s i n the s t r o n g l y complete f i l t e r of the s o c a l l e d G o l d i e Torsion Theory [ 8 ] , where the t o r s i o n submodule of any M e M i s given by Z 2 (M) = T T - 1 ( Z (M/Z^M) ) w i t h TJ : M > M/Z^M c a n o n i c a l . E q u i v a l e n t l y , (1.16) Z 2(M) = { m e M | (Z M:m) i s e s s e n t i a l l e f t i d e a l i n R} . C l e a r l y a l l quotients M/N of M by an e s s e n t i a l submodule N are Z ^ - t o r s i o n . An R-module i s Z ^ - t o r s i o n - f r e e i f and only i f i t has zero s i n g u l a r submodule. Notice that f o r any M e M ^ (M) i s e s s e n t i a l over Z^ (M) . In f a c t Z^(M) i s the maximal e s s e n t i a l extension of Z^(M) i n M i n the sense that i f N i s a.submodule of M which i s e s s e n t i a l over Z (M), then N £ Z 2 (M) . Left" ideal'S'L satisfying*'i'±-)K'of -Lemma" (i s-;i50" were' cald/ed*" • "maxi" i n . [6 ] , In s e c t i o n 5 we w i l l be able to gi v e another c h a r a c t e r -i z a t i o n of these maxi i d e a l s as being " Z^(R)-dense " . Next l e t us s h r i n k the set of e s s e n t i a l l e f t i d e a l s i n R. (1.17) Lemma: For a left ideal D of R, the following conditions are equivalent: i ) \/ 0 7^  r ^ e R and y r ^ e R there exists r e R such that r r ^ ^ 0 and r r ^ e D i i ) for any r e R there is no 0 ^  s e R such that (D:r)s = 0. . Proof: i ) => i i ) C o n d i t i o n i ) says that f o r any r e R and 0 f s e R there e x i s t s x.e (D:r) such t h a t xs f 0 . This i s e x a c t l y c o n d i t i o n i i ) . i i ) => i ) Take any r e R . Hi en saying t h a t f o r every 0 ^  r.. e R 13. ( D : r ^ ) r ^ i- 0 means there i s an r e R such t h a t r r ^ =f 0 and r r ^ e D , as r e q u i r e d . J Any l e f t i d e a l s a t i s f y i n g the c o n d i t i o n s of the Lemma must be e s s e n t i a l . C o n d i t i o n i ) i s the same as the c o n d i t i o n of [16, Prop 4,p.96] and c h a r a c t e r i z e s the dense l e f t i d e a l s which do form a s t r o n g l y complete f i l t e r and l e a d to the complete r i n g of l e f t q u o t i e n t s of R introduced by Utumi. The t o r s i o n theory corresponding to the dense l e f t i d e a l s w i l l be r e f e r r e d to as the Lambek To r s i o n Theory. L e f t i d e a l s D s a t i s f y i n g i i ) of Lemma (1.17) were c a l l e d " s t r o n g " i n [6] where the f o l l o w i n g were proven to be e q u i v a l e n t : (1.18) i ) Z X(R) = 0 i i ) e s s e n t i a l <=> dense i i i ) e s s e n t i a l <=> maxi i v ) dense <=> maxi Thus i f a r i n g has zero s i n g u l a r i d e a l , the e s s e n t i a l , dense, and maxi l e f t i d e a l s a l l c o i n c i d e , and the e s s e n t i a l s form a s t r o n g l y complete f i l t e r [7,Lemme l,p.416]. On the other hand suppose the e s s e n t i a l l e f t i d e a l s i n a r i n g R already form a s t r o n g l y complete f i l t e r . Let M be any R-module. I f m e Z^(M) then there i s an e s s e n t i a l E such that Em C Z^ (M) . So (x) (x) f o r each x e E there i s an e s s e n t i a l E w i t h E xm = 0 ; (x) i e . E C (0:xm) = ((0:m):x) . This says ((0:m):x) i s e s s e n t i a l for- every x e E , which under the hypothis i m p l i e s (0:m) i s e s s e n t i a l . Hence m e Z^(M) and we have Z^(M) = Z^(M) f o r a l l M e M. This means that the idempotent k e r n e l f u n c t o r induced by the e s s e n t i a l s under t h i s hypothis i s e x a c t l y Z^ . From the G a b r i e l correspondence we 14. conclude that the e s s e n t i a l s c o i n c i d e w i t h the maxi l e f t i d e a l s , and so by (1,18) e s s e n t i a l <=> dense and R has zero s i n g u l a r i d e a l . In p a r t i c u l a r , t h i s i s t r u e f o r any semiprime G o l d i e r i n g R [11] ; here the m u l t i p l i c a t i v e l y c l o s e d system ,S C R c o n s i s t i n g of a l l non-zero d i v i s o r s has the common l e f t m u l t i p l e property (1.13) and the set of e s s e n t i a l s i s p r e c i s e l y the s t r o n g l y complete f i l t e r a s s o c i a t e d w i t h S as i n Example (1.12). C o l l e c t i n g a few of the above f a c t s , we have (1.19) P r o p o s i t i o n : For any ring R, the following are equivalent: i ) if 1 is a left ideal in R such that (I:x) is essential for every x e E with E essential, then I is essential i i ) R has zero singular ideal 15. 2. KERNEL FUNCTORS A l l k e r n e l f u n c t o r s are c e r t a i n l y not idempotent. f o r example take a commutative r i n g A ( always w i t h 1 ) and a e A such t h a t a i1 0 , a 1 . Define f o r any A-module M a submodule a (M) = { m e M I am = 0 } a ' C l e a r l y i s a k e r n e l f u n c t o r , but we do not have to look very f a r f o r a r i n g i n which such an a i s not idempotent. In f a c t , . t a k e A = 1 n > 1. Let G = 3?/n32. Then a n ( G ) i - s a proper subgroup of G s i n c e n 2 e a (G) x<rhereas n i a (G) . But a (G/a G) T 0 because n n n n 0 ^ n + a (G) i s a member, n With the aim of l o c a l i z a t i o n i n mind, the main i n t e r e s t i n k e r n e l 1 functors'' is-' to- determine- whether-or' not the- ones- bha-t- arise-*-n a t u r a l l y are indeed idempotent. The set KF(R) has an obious p a r t i a l o r d e r i n g given by (2.1) a $ p <=> a(M ) C p (M) f o r every M e M There i s a s m a l l e s t and a l a r g e s t member w i t h respect to t h i s p a r t i a l o r d e r i n g : namely 0 such that 0 (M) = 0 f o r every M and 0 0 such that °°(M) = M f o r every M. r e s p e c t i v e l y . C l e a r l y O,00 e IKF(R). These are the t r i v i a l t o r s i o n t h e o r i e s which e x i s t f o r any r i n g . (2.2) Example: The G o l d i e T o r s i o n Theory i s the s m a l l e s t n o n - t r i v i a l t o r s i o n theory f o r which a l l modules of the form R/E are t o r s i o n , where E i s any e s s e n t i a l l e f t i d e a l i n the r i n g R. To see t h i s , suppose a e IKF(R) such that a(R/E) = R/E f o r every e s s e n t i a l E. This means that F contains a l l the e s s e n t i a l s . Let L be a maxi l e f t -a i d e a l i n R. Then there e x i s t s an e s s e n t i a l l e f t i d e a l E such that .16. (L:x) i s e s s e n t i a l f o r every x e E. Now E e s s e n t i a l i m p l i e s E e and a e IKF(R) i m p l i e s L e F^ . Hence F^ a l s o c o n t a i n s a l l the maxi' l e f t i d e a l s and so a , (2.3) Example: The Lambek T o r s i o n Theory i s the l a r g e s t n o n - t r i v i a l t o r s i o n theory f o r which the r i n g R i s . t o r s i o n - f r e e . To prove t h i s , l e t p_ be the set of dense l e f t i d e a l s i n R. C l e a r l y R i s t o r s i o n -f r e e w i t h respect to the Lambek To r s i o n Theory s i n c e D e D i m p l i e s (D:l) = D has no r i g h t a n n i h i l a t o r s by Lemma (1.17). Now suppose a e IKF(R) such that a(R) = 0. I f U e F , then (U:r) e F f o r —a —a any r e R. Hence (U:r)s = 0 i m p l i e s s = 0 f o r any r,s e R , which shows by Lemma (1.17) that F^ Q D_ . Any a e KF(R) s a t i s f i e s a 2 = a , so already appears to be " idempotent In order to j u s t i f y our terminology f o r idempotent k e r n e l f u n c t o r s , we in t r o d u c e a product on f i l t e r s ( f o l l o w i n g [2] ). Let F , F be two f i l t e r s of l e f t i d e a l s i n R. Define: -a ' -p (2.4) F *F = { I e l e f t i d e a l s of R I 3 J e F w i t h a ( J / l ) = 0 } . -a -p 1 _J -p F^&Fp i s a f i l t e r , t h e r e f o r e has a uniquely a s s o c i a t e d k e r n e l f u n c t o r which we denote by a*p i n order to w r i t e F *F = F .An R-module• y -a -p a*p M i s a * p - t o r s i o n i f and only i f M has a o - t o r s i o n submodule M' such that M/M' i s p - t o r s i o n . This s t a r product i s a s s o c i a t i v e . From the d e f i n i t i o n of s t r o n g l y complete f i l t e r i t i s c l e a r that . F i s s t r o n g l y complete i f and only i f f o r l e f t i d e a l s I C J i n R such that J e F and J / I a - t o r s i o n we have that I c F . Now Goldman [9,Thm 2.5] st a t e s that a z KF(R) i s idempotent i f and only i f a*o = a • Another way of forming new k e r n e l f u n c t o r s from given ones 17. i s by i n t e r s e c t i o n . More p r e c i s e l y , l e t { o, j i E I } be a f a m i l y of k e r n e l f u n c t o r s i n KF(R). Define a = i n f { a | i e I } by a(M) = Q a ±(M) f o r any M e M . T r i v i a l l y a e KF(R) , a $ a ± f o r a l l i e I , and i f p £ o\ f o r a l l i e I then p < a Furthermore i t i s t r i v i a l t hat i f a = i n f { a . I i e I } then F = n F • -a ' ' -a . 1 This concept of i n f gives r i s e to a c l o s u r e o p e r a t i o n C : KF(R) — > IKF(R) defined by ( 2 .5 ) a i—> a c = i n f { p e IKF(R) | a s< p } More e x p l i c i t l y a°(M) = f) { N g M submodule | a(M/N) = 0 } . c This d e f i n i t i o n does indeed make a an idempotent k e r n e l f u n c t o r by [ 9 ,Prop. 1.1 and Thm. 1.6J and a e KF(R) i s idempotent i f and only i f cr = a*" . V i a the G a b r i e l correspondence, the mapping a i—>• 0 extends to a c l o s u r e o p e r a t i o n on f i l t e r s : ( 2 .6 ) F = F i—y F c = F ° v ' — -a -a — and on s t r o n g l y complete a d d i t i v e c l a s s e s : (2.7) S = S 1—• S c = S° — —o —o — Given any subclass of ^ M , there i s c e r t a i n l y at l e a s t one a e KF(R) f o r which every R-module i n C_ i s o - t o r s i o n , namely a = 00 . F i n d i n g the sma l l e s t such i s eq u i v a l e n t to f i n d i n g the s m a l l e s t s t r o n g l y complete a d d i t i v e c l a s s c o n t a i n i n g C_, Of course we o b t a i n t h i s c l a s s by i n t e r s e c t i o n , and the d e s i r e d k e r n e l f u n c t o r i s i n f { a e -KF(R) | C, } . To o b t a i n the sma l l e s t idempotent one, simply apply the c l o s u r e o p e r a t i o n , and thereby a l s o o b t a i n the sm a l l e s t s t r o n g l y complete Serre c l a s s c o n t a i n i n g C_. This was e x a c t l y the s i t u a t i o n f o r the Goldie T o r s i o n Theory where we cl o s e d the c l a s s { R/E | E e s s e n t i a l l e f t i d e a l i n R } of R-modules. 18. We t u r n to another concept that w i l l be u s e f u l l a t e r . (2.8) D e f i n i t i o n : A k e r n e l f u n c t o r a e KF(R) i s c a l l e d n o e t h e r i a n [9] i f f o r every ascending c h a i n 1^  C ^ C ••. of l e f t i d e a l s i n R whose union i s i n F ,1 e F f o r some n. -a ' n -a In p a r t i c u l a r , every k e r n e l f u n c t o r i n KF(R) f o r a l e f t n o etherian r i n g R enjoys t h i s property. We s h a l l i n v e s t i g a t e the behavior of noetherian k e r n e l f u n c t o r s w i t h respect to the formation of i n f . (2.9) Observation: I f p^,...,P n are f i n i t e l y many noetherian k e r n e l f u n c t o r s , then p = i n f { p, | i = 1 n } i s a l s o n o e t h e r i a n . The proof f o l l o w s immediately from the remark that some member of any acsending c h a i n of l e f t i d e a l s whose union i s in - F must be i n every F . This i s however not the case f o r i n f i n i t e l y many memners of KF(R) P i for' arbitrary*"RT"A"co'unterexampl'e" w i T l " b'e*'given-'in-. (2". 13)' ' a f t e r " the'" f o l l o w i n g d i s c u s s i o n which helps us to determine when a k e r n e l f u n c t o r i s n o e t h e rian. (2.10) D e f i n i t i o n : A f i l t e r I? i s s a i d to have a co f i n a l subset of  f i n i t e l y generated l e f t i d e a l s i f every U e F_ conta i n s a f i n i t e l y generated l e f t i d e a l which i s a l s o i n J_ . ' (2.11) Observation: I f a f i l t e r F^ has a c o f i n a l subset of f i n i t e l y generated l e f t i d e a l s then a i s no e t h e r i a n . To see t h i s , l e t 1^  C ' ' * b e any ascending c h a i n of l e f t i d e a l s i n R such that U = I J I i s i n F^ . Then U contains a f i n i t e l y generated l e f t i d e a l (x,,...,x ) e F , and so some I. must c o n t a i n a l l the x..,...,x . 1 n y -a ' 2. I n This says that some I. e F J i -a The converse of t h i s i s not yet c l e a r as remarked i n [9]. However we make the f o l l o w i n g : 19. (2.12) Observation: Suppose every U e F^ i s at most countably generated. I f a i s noetherian then F^ has a c o f i n a l subset of f i n i t e l y generated l e f t i d e a l s . The proof i s obtained simply by t a k i n g chains where we keep adding on generators of l e f t i d e a l s i n F^ . (2.13) Example: Let A = k[x^,X2>...] be the commutative polynomial r i n g i n i n f i n i t e l y many indeterminants over a f i e l d k. Let 2 3 S. = { l , x . , x . ,x. ,...} be the m u l t i p l i c a t i v e l y c l o s e d system i n A x x x x defined by x^ , and l e t a be the idempotent k e r n e l f u n c t o r a s s o c i a t e d w i t h S^ as i n Example (1.12) . Since A i s commutative, f o r any M e m e a.(M) <=> x.nm = 0 f o r some n , and F c o n s i s t s of those l x -o. x i d e a l s i n A that meet S. . This means F has a c o f i n a l subst of x -a. x f i n i t e l y generated i d e a l s of the form As , s e S^ and so by (2.11) each,..a.. iss-noe-thaE-Aan!.*.-Leta,-•=•. inf..•.{-,ov.. I. it-=«•!-.,*..... ,«>•< } C o n s i d e r - . • the c h a i n of i d e a l s (2.14) ( x 1 ) C (x 1,.x 2) C ( x 1 5 x 2 , x 3 ) C ... and l e t U be t h e i r union. Since a^(A/U) = A/U f o r every i , U >e F^ . But a(A/(x^,...,x^)) ^ A/(x^,...,x n) f o r any n because i f m > n , x + (x, ,...,x ) cannot be a a - t o r s i o n element and consequently m i n m -i J cannot be a a - t o r s i o n element. Hence no member of t h e chain (2.14) i s i n F , and' a i s not noetherian. -a ' (2.15) D e f i n i t i o n : A f i l t e r F_ i s c a l l e d m u l t i p l i c a t i v e i f U,V e F_ i m p l i e s UV e F_ . (2.16) Remark: A s t r o n g l y complete f i l t e r i s always m u l t i p l i c a t i v e s i n c e U C (UV:v) e F_ f o r every v e V . This i s the same as saying F^ i s m u l t i p l i c a t i v e f o r any a e IKF(R). The converse of t h i s remark i s however not tr u e i n gen e r a l . The f o l l o w i n g counterexample i s 20. i n d i c a t e d i n [2,p.158]. (2.17) Example: Let k [ x ^ , x 2 , . . . ] be the commutative polynomial r i n g i n i n f i n i t e l y many indeterminants over a f i e l d k . Let (x^x^.)^^. be the i d e a l generated by a l l the x - j _ x j a n c* P u t A = k [ x 1 , x 2 , . . . ] / ( x i x _ . ) i ^ j CO Let = x, + ( x . x . ) . ,. and I = (5, ), , be the i d e a l i n A k Ic i j XT*J k k=l generated by a l l the 5^ . Consider the set F_ of i d e a l s i n A c o n t a i n i n g a power of I . C l e a r l y F i s a m u l t i p l i c a t i v e f i l t e r of i d e a l s i n A. Denote the corresponding k e r n e l f u n c t o r by T such that I? = F . i "° i Let J = ( £ . ) . ., be the i d e a l i n A generated by a l l the £. l 1=1 o J X t h s ( i powers of the S^'s )• Then f o r any a e I , I a CI J f o r g l a r g e enough s. So I C (J:a) e F^ f o r every a e I ( s depending complete, and we conclude that x i s not idempotent. By imposing some r e s t r i c t i o n s we do get a p a r t i a l converse to Remark (2.16) i n the case of commutative r i n g s . (2.18) P r o p o s i t i o n : Let A be a commutative ring. If a filter F has a cofinal subset of finitely generated ideals, then F is multiplicative if and'only if the associated'kernel- functor- a- is idempotent-. Proof: I f a i s idempotent, the c o n c l u s i o n f o l l o w s from Remark (2.16). Conversely, l e t M be any A-module. For any 6 = m + o(M) e a(M/o(M)) there i s a f i n i t e l y generated i d e a l I = (a.,...,a ) e F such that J b 1 n -a 10 = 0 ; i e . a.m e a(M) f o r i = l , . . . , n . Then f o r each i there i s a i U. e F w i t h U.a.m = 0 . Let J = f ) U . e F . Then Jim = 0 and x -a i i i -J l e F^ so that m e a(M) and a(M/o(M)) = 0 . I 21. (2.19) Corollary: If F_ is a filter of ideals in a commutative ring A then the following assertions are equivalent: i) F_ is multiplicative i i ) J_ i s strongly complete i i i ) the associated kernel functor is Idempotent , I 3. LOCALIZATION FUNCTORS In t h i s s e c t i o n a l o c a l i z a t i o n f u n c t o r i s co n s t r u c t e d f o r each a e IKF(R) and some of the b a s i c p r o p e r t i e s are obtained. The f o l l o w i n g theorem c h a r a c t e r i s e s a c l a s s of modules which turns out to be q u i t e important. (3.1) Theorem: For any a e IKF(R) with torsion theory ( L . S ,V ) and A e M the following are equivalent in M : i ) i f M/N e S^ for N Q M submodule then any R-map N > A extends to an R-map M • A / i e . M(M,A) >- M(N,A) > 0 i s exact whenever N £ M and M/N e S -a i i ) I(A)/A i s a-torsion-free ; ie. I(A)/A e i i i ) i f U e F^ and g : U > A is any R-map then there exists a £ A such that g(u) = ua for every u e U i v ) i f E e F^ i s an essential left ideal in R , then any R-map E >• A extends to R v) any R-map U > A with U e F^ extends to R ; i e . for every U e F a M(RjA) > M(U,A) > 0 is exact v i ) E x t R 1 ( S , A ) = 0 f o r every S e S v i i ) f o r any essential extension M of N with (A:m) e F^ for every m e M any R-map N >• A extends to an R-map M >• A Proof of equivalence of these c o n d i t i o n s can be found s c a t t e r e d through the l i t e r a t u r e : i ) <=> v) by [9,Prop 3.2]; i ) <=> i i ) <=> i i i ) ' by [17,Prop 0.5] where such an A e M i s c a l l e d d i v i s i b l e ; i v ) <=> v) by [32,Thm 11]; i ) <=> i i ) <=> v) <=> v i ) by [33,Prop 2.4]; i ) <=> v i i ) by [20,Prop 1.2] 1 23. (3.2) Definition: An R-module A is called o-injective [9] i f A satisfies any ( hence a l l ) of the conditions of Theorem (3.1). If in addition the extension in i) in the Theorem is unique , ie. M(M,A) - M(N,A) for N £ M and M/H £ S F L , then A is called f a i t h f u l l y a-in;]ective. From [9,Prop 3.1] we have (3.3) Proposition: A e M is faithfully a-injective if and only if A is a-injective and A e V Q . For any a e IKF(R) l e t A q denote the f u l l subcategory of M consisting of the f a i t h f u l l y a-injective R-modules. Regarding R— as a f u l l subcategory of the above Proposition (3.3) gives an- embedding* 'functor'" J- :• A- >----V- (•• subscripts- have-been'-dr-opped-*-as we are considering a fixed for now ). Composing with the embedding K : V > M we get the embedding (3.4) E : A - i - V M For C e V l e t D(C) C. 1(C) be the extension of a(I(C)/C) by C ( ie. 0 > C >• D(C) > a(I(C)/C) > 0 exact in 0M . ) K— such that D(C)/C = a(I(C)/C) and CCD(C) . (3.5) Proposition: For any C £ V_ we have D(C) e A . Proof: As C C D(C) £ 1(C) , D(C) e V by (1.8)ff. Since 1(C) is clearly an essential extension of D(C) ( as i t is already essential over C ) 1(C) is the injective hull of D(C) by [16,Prop 10,p.92]; ie. I(D(C)) = 1(C) . Now I(C)/D(C) = (I(C)/C)/(D(C)/C) = (I(C)/C)/a(I(C)/C) is a-torsion-free because a is idempotent. Hence D(C) is a-injective 24. by Theorem (3.1,ii), and together with Proposition (3.3) this means D(C) 6 A . ' I Here i t should be remarked that monomorphisms in A and .in V_ coincide, and hence also coincide with monomorphisms in M . Now D : y_ > A is easily made into a functor: since D(B)/B = o(I(B)/B) is a-torsion, for any f : B y C in V there is a unique D(f) by the fa i t h f u l o-injectivity of D(C) that makes the diagram B - > D(B) (3.6) I D(f) i 4-C y D(C) commute where i , i are inclusion. If f in (3.6) is a monomorphism, b 6 B such that D(f)(i (b)) = 0 , then i_f(b) = 0 and so b = 0 B O since i f is a mono. Hence ker D(f) f\ B = 0 . But D(B) essential over B implies ker D(f) = 0 . This shows that D preserves monomorphisms. Furthermore, for any A e A , C e V_ with i : C y D(C) inclusion (3.7) i * : A(DC,A) — y V(C,JA) is clearly a natural isomorphism. Composing with the adjoint of Proposition (1.10) and putting Q = DF we have that Q is l e f t adjoint to the embedding functor. E . Since both F and D preserve monos, [22,Prop 5.1,p.129] again gives us that A is an abelian category, and by [22,Prop 12.1,p.67] Q is exact. Notice that A is not in general an abelian subcategory of M as the embedding functor E need not be exact. Since the category M is complete, [22,Prop 5.1',p. 129] 25. gives us that A is complete. Collecting these results ( see also [17,Prop 0.8] ) yields the following; (3.8) Theorem: For any a e IKF(R) , the full subcategory A consisting of the faithfully a-injective R-modules is a coreflective subcategory of M with coreflector Q = DF . Furthermore A is a complete abelian category and the coreflector Q is exact. (3.9) Definition: The endofunctor EQ : M > M w i l l be called K- R— the localization functor relative to a e IKF(R). In effect, the functor EQ provides a a-injective hull for every M e M as in [33,Prop 4.2] and can be computed by: This Proposition (3.10) in the particular case M = R appears as [27,Prop 1.7]. More information regarding this localization can be obtained by investigating the unit and counit of the adjunction n : Q —{ E . (3.10) Proposition: For any M e M EQ(M) = { x e I(M/cr(M)) j (M/a(M):x) e F } , Proof: Immediate by our definition of D which i s EQ(M)/F(M) = EDF(M)/F(M) = a (I (FM)/F (M) ) and that a(M) = { m e M | (0:m) e F } . I The unit is given by cj> = n (1 Q(M) ) : M—> EQ (M) and i s easily computed to be: M * EQ (M) (3.11) M/a (M) Hence there is a natural mapping <j>M : M »- EQ (M) for every M e whose kernel is exactly a(M) . ( 3 . 1 2 ) Proposition: An R-module M' i s faithfully a-injective if and only if EQ(M) = M . Proof: If M = EQ(M) , then M is f a i t h f u l l y a-injective by Proposition ( 3 . 5 ) . Conversely, M f a i t h f u l l y a-injective implies a(M) = 0 , so that <}>^  : M >• EQ(M) is inclusion. Since EQ(M) is already f a i t h f u l l y a-injective and EQ(M)/M is a-torsion, there exists a unique a such that the diagram M : > EQ(M) ( 3 . 1 3 ) ~ / 1 EQ(M) (uniquely) commutes From <|> «:1 • = 1^/wv •<!>,, = fy.'Ct'fy.. we have <J> *a = 1 _ b y M M EQ(M) M M M M EQ(M) uniqueness, which together with  a'$n ~ 1^ implies that the inclusion <J> is an isomorphism. Hence EQ(M) = M . J ( 3 . 1 4 ) Corollary: i ) (EQ) 2 = EQ i i ) the counit of the adjunction n : Q — \ E is the identity. Proof: i ) EQ(M) is f a i t h f u l l y a-injective. i i ) for any A e A the counit of r\ is given by n _ 1 ( l E A ) : QE(A) • A But EQE(A) = E(A) and E being an embedding implies QE(A) = A Furthermore 1 works!^so we have i t by uniqueness. I We write the counit of n as 1 : QE *• Id^ Here we can mention a few more simple facts: (3.15) Observation: M is a-torsion <=> Q(M) = 0 <=> EQ(M) = 0 . ( => ) by construction of the functor ( <= ) by the fact that a(M) = ker $ (3.16) Observation: If U e F_a then Q(U) - Q(R) and hence EQ(U) = EQ(R) . This is because 0 y U y R y R/U y 0 i s exact in M , consequently 0 • Q(U) —> Q(R) y Q(R/U) > 0 is exact in A . But R/U is a-torsion, so Q(R/U) = 0 and we have Q(U) - Q(R) in A . Since E preserves isomorphisms ( always ! ) EQ(U) - EQ(R) in M . The objective of such localization is to study the ring R by means"" of"* EQ-OR.')"" whi'cir--shoul'd"aiso*-b-e"a"ring'; In1-order forgive-' EQ(R) a suitable ring structure, notice that as A is an abelian category on i t s own right, the adjunction n : A(Q(M),A) - M(M,E(A)) is an isomorphism of abelian groups. Putting M = R there is an isomorphism E(A) - A(Q(R),A) of abelian groups which says that E is representable ( in the sense of [17] ). It is clear from the general theory of representable functors that ( Q(R), 0(1) ) is a representing pair for E , where we write (j> = $ for the canonical map R y EQ(R) . The rest of the story now follows immediately from [17,Prop 1.1] which is stated here in i t s entire generality. (3.17) Proposition: ( Lambek ) If C_ is an additive category, U : C y M a representable functor with representing pair R— ( A , s ) , s s U ( A ) , such that U(C) - C(A ,C) is an abelian o o o o o group isomorphism for every C e , then the following are true: 28, i) ^(^Q)  c a n ^ e ma-de into a ring S - C ( A Q , A O ) with underlying abelian group same as that of U ( A Q ) i i ) the map R U ( A Q ) given by r I—> r s Q i s a ring homomorphism 3 •: R — y S i i i ) for each C e C U(C) i s a l e f t S-module, call, it T(C) iv) for each f : C > C' i n £ U(f) i s an S-map, c a l l i t T(f) v) T : C_ > M i s a functor such that . U^T = U C where : M >• M i s change of rings functor via 3 - see / f I 2 7 ) o— R— vi) T i s representable with representing pair ( A , .s ) and S >• T(A ) by x i >• xs^ i s an isomorphism. From this proposition i t now follows that EQ(R) = Q(R) is a ring with unit ^(1) such that Q(R) A(Q(R),Q(R)) and <p : R > EQ(R) is a ring homomorphism such that the induced R-structure by change of rings coincides with the original structure as in [9,Thm 4.1]. There is a functor T : A—-s» , .M such that hQCR;-4 U. T = E and T_ * A(Q(R),_?) giving each fa i t h f u l l y cr-injective R-module an EQ(R)-module structure ( see also [9,Cor 4.2] ). In Lambeck's terminology [17], ( Q(R), T ) is called the completion of ( R, E ) . Putting together the facts thus far, we have the following commutative diagram of categories and functors: M EQ(R)-A(EQ(R) ,?) 29. It should be pointed out here that this functor Q cannot be considered the same as Goldman's localization functor Q which is actually EQ . This is most dramatically illustrated by the fact that Q is exact while EQ is not in general right exact. It w i l l be easy to give a counterexample in Section 4 after a discussion of the important "property T", but the basic reason is that A need not be an abelian subcategory of M . In fact, i t turns out to be an abelian subcategory i f and only i f EQ is exact - see [33,Thm 3.13]. Since the endofunctor EQ arose from a pair of adjoints Q — I E , i t gives rise to a monad [25] which by Corollary (3.14) is particularly simple. E x p l i c i t l y i t is given by the commutative diagrams':* (3.19) written ( EQ, <j>, \ ) where <j> : Id^ >• EQ is the unit of the — 2 adjunction n : Q — I E and i = E1Q : (EQ) >• EQ i s the identity natural transformation since the counit. 1 of n is the identity and (EQ)2' = EQ . Having Theorem (3.8) at our disposal, we can apply [25,ThnT 2,p.75] to conclude that E is a monadic functor ( caution: our coreflective as in [22] corresponds to reflexive in [25] ). This means the following: the endofunctor EQ gives rise to a category M of socalled EQ algebras. The objects of M are pairs ( A, a ) where A e M and a : EQ(A) >- A . is a morphism in M such that the following diagrams commute: 30. (3.20) (EQ)2(A) -M(«l> EQ (A) A EQ (A) -> A Morphisms from ( A,"a ) to ( B, 3 ) are morphisms f : A in M such that the diagram EQ(A) E Q ( f ) > EQ(B) (3.21) 3 A -> B commutes. We get a functor L : A >• M*"* by defining L(A) = ( E(A) , E(± A) ) „EQ EQon objects and the obv/ous one on morphisms. That M_7X is indeed a category and L a functor has been shown in great generality by Pareigis [25,Thm l,p.62J. Now E monadic means that L : A > M" is an isomorphism of~categories. Furthermore by Theorem" ('3T8')'", EQ M i s abelian. Hence we can regard the category A as an abelian category of EQ algebras. This was done by Heinicke [10,Thm 4.3] under somewhat more general circumstances using the Eilenberg-Moore construction directly. In [10], localization functors are defined via natural transformations and made to correspond with certain monads deemed localizing. However, the functor EQ and i t s monad ( EQ, <j>., i ) are the canonical choices as seen by [10,Thm 2.4]. Several other equivalent descriptions of this localization process can be found in the literature. In [9] Goldman gives an explicit elementwise characterization of EQ(M) which for any o-torsion-free R-module C is essentially given by EQ(C) = M(U,C) / = UeF -a where = is an equivalence relation such that f e M(U,C) and 31. g e M(V,C) are related when there exists some W e F with WC D(°| V on which f and g agree. We can translate this description of EQ into (3.22) EQ(M) = lim M(U,M/a(M)) for any M e M UeF -a by making the observation that inclusion in F induces a direct system of the abelian hom-groups with the required R-module structure defined on the direct limit by letting r [ f] be the equivalence class of (U:r) > M x \ > f (xr) for any r e R , and any equivalence class [f] , f e M(U,M). This coincides with the construction in [2] as well as that in [7,33] using the quotient category of M with respect to the. strongly complete Serre class S 32. 4. PROPERTY T Let EQ be the localization functor "relative to a e IKF(R) and <|> the unit of the adjunction n : Q — I E . Denote the localised ring EQ(R) by Q ( a l l other notations w i l l be the same as in the previous section ). (4.1) Lemma: There is a natural transformation K : Q 8 ? > EQ of endofunctors on M given by K^(q@m) = q(c|>^ (m)) for any M e M and m e M , q e Q . Proof: The set map Q x M • EQ(M) (q,m) > q(<j)M(m)} is bilinear and R-balanced for any R-module M , thus extends uniquely to an R-map K : Of 0 R M':—> EQ"(M)*! by the universal property of the tensor product. Let f : M > N be any R-map and consider the diagram: • ' *M KM M > EQ(M) < Q 8 M (4.2) f N > EQ(N) -f Q 8 N Now for any generator q,@m. e Q. © M. EQ(f)-KM(q0m) = EQ (f ) (q (m)) ) = q(EQ(f).4»M(m)) = q(<|) *f (m)) by naturality of KN-(10f)(qOm) = tcN(q0f(m)) = q(* N-f (m)) Hence extending linearly, we have naturality of K . 1' EQ(f) 18f 33. (4.3) Lemma: If Q(f>(I) = Q then I e F . Proof: Exactness of I >• R >• R / l > 0 implies the exactness of Q@I Q8R > Q8R/I >• 0 . The image of a is Q4> (I) . But Q<f>(I) = Q - Q®R . Thus Q0R/I = 0 , which implies KR^ = 0 and so ~ 0 • Hence R / l = ker 0 Ryj = o"(R/l) which implies that I e F .' I -a (4.4) Lemma: Q 6 (R/U) i s a-torsion for every U e F R —a Proof is given in [27,Lemme 1.3] The three Lemmas above provide some information about the general case of localizing relative to an arbitrary idempotent kernel functor. They also raise the following questions: i) - when-is" fhe'-na-turai'- tran-sformati-on*'in,'"Lemma' -(4-.'i)' a" rcaturai'"isomor-phism-?'-i i ) when is the converse of Lemma (4.3) true ? i i i ) when is Q 0 (R/U) = 0 ( ie. "really" torsion ) for every U R . in the strongly complete f i l t e r 1? . We shall see from the answers in Theorem (4.5) below that these questions are intimately related. It i s also of interest to know when the functor A(Q(R),_?) : A >- QM ( see diagram 3.18 ) is a natural isomorphism. For i f this is the case, every X e is isomorphic to A(Q(R),A) for some A e A . Then regarding X as an R-module by change of rings via $ , U* (X) - U<f>A(Q(R),A) = E(A) is f a i t h f u l l y a-injective. On the other hand, i f for every X e , (X) is fai t h f u l l y a-injective, then QU(j)A(Q(R) ,_?) = I d A and A(Q(R),QU I) = Id by diagram (3.18) which implies that Q-A(Q(R),?) is a natural isomorphism . 34. The following Theorem establishes the connection between the foregoing remark and the above questions. (4.5) Theorem: For any a e IKF(R) , the following statements are equivalent: i ) the functor A(Q(R) ,_?.) : A >• is a natural isomorphism i i ) (X) is a-torsion-free for every X e i i i ) Q<f>(U) = Q for every U e F^ iv) (X) is faithfully a-injective for every X e v) K : Q @ ? > EQ i s a natural isomorphism. R— vi) the functor EQ i s right exact and commutes with direct sums v i i ) Q 8R(R/U) = 0 for every U e F^ v i i i ) the functor EQ is right exact and F^ has a cofinal subset of finitely generated left ideals ix) the functor EQ preserves colimits . Proof: The i n i t i a l remark establishes i) <=> iv) ; vi) <=> ix) i s an immediate consequence of the dual statement to [22,Cor 6.3,p.55] since DM i s a conormal category with direct sums. In fact ix) is is.— equivalent to the weaker statement: the functor EQ preserves direct limits ; i i ) <=> i i i ) <=> iv) <=> v) <=> vi) by [9,Thm 4.3] with naturality following by Lemma (4.1) ; v i i ) <=> v) <=> i i i ) <=> i i ) by [33,Thm 3.2] ; v i i i ) => i ) and v i i i ) => v) by [7,Cor 2,p.414] ; v i i ) <=> i i ) <=> v i i i ) <=> v) <=> i ) by [27,Prop 2.8] where a new proof is offered. J (4.6) Definition: We say that a e IKF(R) has property T [9] ( for tensor product ) i f any ( hence a l l ) of the conditions of the above Theorem (4.5) are satisfied. 35. In {9J Goldman proves two more interesting equivences which are useful in determining property T : (4.7) the functor EQ commutes with direct sums if and only if a is noetherian (4.8) the functor EQ i s right exact if and only if every U e i s "<r-projective" in the sense of the following: Definition: An R-module P is called a-projective i f for any epimorphism C >• C" of a-torsion-free modules in M , any R-map P > C" can be l i f t e d to an R-map P' >• C on a submodule P' of P with P/P' o-torsion making the diagram 0 > P' y P C. >• C" > 0, commute. (4.9) Example: Since any projective R-module is clearly a-projective for any a e IKF(R) , i t is evident that for a l e f t noetherian hereditary ring R , every a e IKF(R) has property T . (4.10) Example: If R is a l e f t semisimple artinian ring, then R is ( l e f t ) hereditary by Wedderburn's Theorem [13] and also l e f t noetherian [16,p.69]. Hence by Example (4.9) every a e IKE(R) for a- l e f t semisimpl-e- artinian ring R has property-T . C 4 . l l ) Example: Let S C R be a multiplicatively closed system with associated idempotent kernel functor a as in Example (1.12). Suppose S has the common l e f t multiple property (1.13). Then F^ contains Rs for every s e S and F^ has a cofinal subset of principal l e f t ideals. Let U e F , s e S U , p : C -—> C" an epimorphism of a-torsion-free R-modules, and f : U > C" any R-map. For some c e C , p(c) = f(s) . The R-map Rs > C defined 36. by rs '—y rc makes the diagram 0 y Rs > U f C ^ > C" -> 0 commute Also U/Rs C R/Rs and so U/Rs is a-torsion. This means that every U e F^ is a-projective which by (4.8) implies EQ is right exact. Hence by Theorem (4.5,viii), a has property T . r The converse of the above need not hold in general, i e. the common l e f t multiple property is not a necessary condition for a to have property T . For a counterexample, consider the ring R of 2x2 matricies over a division ring D and the multiplicativly closed system S consisting of matricies of the form (o o) w^ t' 1 0 ^  d e D. Le-t- p« be'>-the*-as<3Qe<L-a*ed--idempotan^  R»- i s — simple artinian, p has property T by Example (4.10), but S does not have the common l e f t multiple property because taking (o o) £ R ' (o o) £ S ' (o o)(o o) = ( dij }2x2 ( J I) implies d = 0. Returning to the general situation of a multiplicatively closed system S in a ring R , the common l e f t multiple property does insure that every <f>(s) for s e S under the canonical map (j) : R >• EQ(R) has a l e f t inverse in EQ(R) . To see this, pick any s e S and define k g : R > R by r /—> rs . This induces the exact sequence k (4.12) 0 y K y R R > R/Rs > 0 where K = ker k g , R/Rs = cok k g . Since a has property T , we have an exact sequence EQ(R) -^y EQ (R) >• EQ (R/Rs) y 0 where p = EQ(kg) is the unique extension by f a i t h f u l a-injectivity 37. of EQ(R) defined by qi—>- q (s) . Now R/Rs e F implies EQ(R/Rs) = 0 and so p is an epimorphism. Hence 1 = q(<Hs)) for some q e EQ(R) and we obtain the desired l e f t inverse. If we demand that each <j>(s) is also to have a right inverse in EQ(R) then rs = 0 implies cj>(r)<f>(s) = <|>(rs) = 0 and consequently <(>(r) = 0 ie. r e a(R) which means there i s some t e S such that tr = 0 . Hence a necessary condition for right i n v e r t i b i l i t y of <)>(s) is : (4.13) i f rs = 0 with r e R then tr = 0 for some t.e S . This condition i s also sufficient because i t implies K C o (R) in (4.12) so EQ(K) = 0 making u = EQ(kg) a monomorphism :( hence an isomorphism ) . Again letting q be a l e f t inverse of <j>(s). > from u(cj>(s)q - 1) = (<(>(s)q ~ l H ( s ) = 0 we have <f>(s)q = 1 and he-nee'.- (^s^ ).-- haS"a-2-sided..-in.v.er.se,.in- . EQ,(»R.)*-.... Wei-hav.e«.thus» ar,r.iv,ed<. . at the f u l l generalization of the classi c a l Ore condition: (4.14) Definition: A multiplicatively closed system S C R is called a l e f t denominator set [4] i f S satisfies both the common l e f t multiple property (1.13) and condition (4.13) above. (4.15) Definition: Let S C R be a multiplicatively closed system. A ring of l e f t fractions for R with denominators in S [2,7] is a pair ( Q', \p ) where Q" is a ring and' ifi : R *• Q is' a ring' homomorphism satisfying the following three conditions: i) i f i^(r) = 0 then sr = 0 for some s e S i i ) 4>(s) is invertable in Q for every s e S i i i ) every element of Q has the form ^(s) % ( r ) r e R , s e S (4.16) Theorem: For a multiplicativelu closed system S i n a ringr R' with associated idempotent kernel functor a the following are 38. equivalent: i ) R has a ring of left fractions with denominators in S i i ) S is a left denominator set i i i ) for every s e S , $(s) via the canonical <j> : R > EQ(R) is invertable in EQ(R) . Proof: i i ) => i i i ) f o l l o w s from the preceeding d i s c u s s i o n . i i i ) => i ) f o l l o w s from P r o p o s i t i o n (3.10) and the f a c t t h a t every U e F^ meets S . The p a i r ( EQ(R), <j) ) becomes the ( unique up to isomorphism ) r i n g of l e f t f r a c t i o n s f o r R which i n t h i s case i s a l s o the u n i v e r s a l S - i n v e r t i n g o b j e c t i n [4]. i ) => i i ) i s e a s i l y proven as i n [7,Prop 5,p.415]. J I f S contains only the non-zero d i v i s o r s , a r i n g of l e f t f r a c t i o n s w i t h denominators i n S ( i f i t e x i s t s ) i s c a l l e d a c l a s s i c a l r i n g of l e f t q u o t i e n t s f o r R . From the Theorem we can immediately deduce [16,Prop l , p . l 0 9 ] : (4.17) C o r o l l a r y : A ring R has a classical ring of left quotients if and only if it satisfies the classical Ore condition. For a commutative r i n g A ,. every m u l t i p l i c a t i v e l y c l o s e d system S i s a denominator s e t . Therefore the r i n g of f r a c t i o n s of A w i t h denominators i n S always e x i s t s and can be const r u c t e d i n the c l a s s i c a l way as i n [l,Chap 3], Because of property T , t h i s c o n s t r u c t i o n extends to every A-module M by EQ(A) 0^ M e x a c t l y as i n [ l , P r o p 3.5] which gives the module of f r a c t i o n s w i t h denominators i n S . 39. (4.18) Proposition: If a e IKF(R) has property T , then the canonical map (j) : R > EQ(R) is a ring epimorphism and EQ(R) . i s flat as a right R-module. 2 Proof: The isomorphism EQ(R) = (EQ) (R) - EQ(R) Q EQ(R) i s induced by multiplication, so <j> : R > EQ(R) is a ring epimorphism by [28,Prop 1.1], Since EQ - EQ(R) 8 ? is a natural isomorphism, R — EQ(R) is f l a t in J In particular, the results of [28] apply to any a e IKF(R) with property T in which case <j> : R >- EQ(R) is a l e f t localization in the sense of Silver. However, the converse of Proposition (4.18) is not true in general. (4.19) Example: Let A = k[x,y] be the commutative polynomial ring in two indeterminants over a f i e l d k . Let M = (x,y) be the maximal ideal of A , and take J_ to be the f i l t e r consisting of ideals containing a power of M . Since F_ is clearly multiplicative, F_ is a strongly complete f i l t e r by Corollary (2.19). Let p be the associated idempotent kernel functor. Clearly A is p-torsion-free ( a s A is an integral domain ) so the canonical map (j> : A EQ(A) is inclusion.If we l e t K denote the f i e l d of fractions k(x,y) of A we have K = 1(A) and p(K/A) = 0 . Hence A is f a i t h f u l l y p-injective and hence A = EQ(A) . By Observation (3.16) EQ(M) = EQ(A) = A . Consequently EQ(M) is not isomorphic to A @^  M - M and thus p does not have property T . Nevertheless <|> : A —•> EQ(A) = A is a ring epimorphism and A is certainly f l a t as a right A-module. This example shows more: (4.20) Since A is noetherian, p is also noetherian. Hence EQ 40, cannot be right exact relative to this p . But Q is s t i l l exact as always. (4.21) Not every torsion theory for a commutative ring arises from a multiplicatively closed system since a l l these do have property T . (4.22) The product of torsion modules need not be a torsion module. co . n In the present example, ILA/M is not p-torsion ( the element n=l n - i 0 0 { 1 + M Jn_-L cannot be k i l l e d by any single power of M ) but each A/Mn is p-torsion by construction of the f i l t e r . A similar example of this can be found in abelian groups. Consider the group CO 2(p ) ( written additively ) with generators 0 ^ , 0 ^ , • . • , c n , . . . and relations pc^ = 0 , pc^ = c^ P c n +^ = c n ' ' " ' ' ^ & § r o u P s CO ^(p )/Zc_^ i = 1,2,... are a l l torsion in the usual torsion theory for , but their product is not a torsion group. The above Example (4.19) is a concrete version of one indicated in [9,Ex 2,p.45]. Even though the converse of Proposition (4.18) is not true iri general, a partial converse in this connection can be found in the literature. (4 .23) Theorem: A' ring map i|T :* R* *• Q* ' is an eplmorphism' and" Q" is' flat as a right R-module if and only if the set of left ideals I in R such that. Q^.(I) = Q Is a strongly, complete filter and for the localization relative to this torsion theory there is an isomorphism Q - EQ(R) making the diagram R' ~" 'EQ(R) commute Proof i s given in I17,Thm 2.7J and [27,Prop 2.7]. In the light of the above Theorem we may regard the l e f t localizations of Silver [28] as arising relative to an idempotent kernel functor which appears very close to having property T . 42. 5. REPRESENTATIONS OF IDEMPOTENT KERNEL FUNCTORS AND THEIR RELATIVE LOCALIZATIONS -Let cr e IKF(R) be arbitrary and let ( F, S_, V ) be i t s corresponding torsion theory. Again notations from previous sections w i l l be retained. The o-torsion modules _S are generated by the cyclic modules R/U with U e F_ , i n the sense that for any M e _S there is an epimorphism 0 R/U > M UeF The a-torsion-free modules V_ are cogenerated by the modules I(R/I) where I is a l e f t ideal in R such that a(R/I) = 0 ( called closed l e f t ideals in [17] ) } in the sense that for any C e V_ there is a monomorphism C > IT { I(R/I) | a(R/I) = 0 } = V via a factorization through the injective hull of C and then by a construction of Jans ( described in [17,p.6] ) , Let M be any R-module and put k(M) = O { ker(f) [ f e M(M,V) } Clearly a(M) Q k(M) by (1.9) as V is a-torsion-free. On the other hand, for any m e k(M) , M(Rm,V) = 0 because we can always f i l l the diagram 0 —-c Rm M by in j e c t i v i t y of V . Now i f C is any a-torsion-free module, we have a monomorphism C > V . Hence 0 > M(Rm,C) > M(Rm,V) = 0 is exact which implies M(Rm,C) = 0 for every C e V . Consequently 43. by (1.9) Rm f S and therefore m 6 (T(M) . Hence V completely determines the torsion theory as: (5.1) a (M) = ker(f) | f e M(M,V) } By reversing the procedure, i t is clear that any R-module S determines a kernel functor i f we take V = I(S) . We denote the kernel functor which arises i n this way by x and observe that x i s idempotent and in fact is the largest idempotent kernel functor for which S is torsion-free [9,Thm 5.1]. From the above discussion we have: (5.2) Proposition: For every o e IKF(R) there exists M e M such that a = x,, M Notice that' an" R-mo'duTe"" X"" is'"'T;v,-t''6'rsion M <=> M(X,I(M)) = 0 <=> V x £ X \/m e. M r £ R ^- rx = 0 and rm f 0 Clearly the R-module M whose existance was asserted in Proposition (5.2) is not unique - not even up to injective h u l l . But i t is unique up to a relation that is manufactured to do precisely that job. (5.3) Definition: i ) We say that a module M is cogenerated by a module G i f M can be embedded in a product of coppies of G . i i ) Two modules are called similar [22] i f they cogenerate eachother. (5.4) Lemma: Two modules M, N give rise to the same torsion theory ( ie. x = x^ ) if and only if M and N have similar injective hulls. Proof: ( Storrer [17, Appendix] ) We f i r s t remark that a necessary and sufficient condition for a module X to be T -torsion-free i s that X be cogenerated by I(S) . This condition is obiously sufficient because products and submodules of torsion-free monules are again torsion-free. Necessity follows from the fact that M(Rx,I(S)) =f 0 for any 0 ^ x e X since RxC X is x -torsion-free By i n j e c t i v i t y we can f i l l the diagram 0 >• Rx > X y''3' f 0 ' K s f which by the universal property of direct products gives a map X >• I(S) into the X-fold product of I(S) which must be mono. Now i f I(M) amd i(N) are similar, then V = V TM TN • because cocenerating is transitive, and hence x — x M N Conversely i f x w = x„ then M is x -torsion-free and we M N N have an embedding e : M > I(N)^ for some J-fold product of I(N) . By i n j e c t i v i t y we can f i l l the diagram 0 > M >• I(M) /* K N ) J ^ with a mono since I(M) is essential over M . Therefore I(M) is cogenerated by I(N) . By symmetry, I(N) is cogenerated by I(M) and hence M and N have similar injective hulls. \ (5.5) Proposition: An idempotent kernel functor a ~ Xg has property T if and only if the l o c a l i z a t i o n EQ(R) of the ring r e l a t i v e to a is f l a t as a right R-module and I(S) is s i m i l a r to the i n j e c t i v e left R-module i W = • M.(EQ(R) ,<Q/2) . 45. Proof: If a has property T , i t has already been shown in Proposition (4.18) that EQ(R) is f l a t as a right R-module. It remains to show a = xTT . Now _M(M,W) - M(EQ(R) 0_ M,Q/2) - JJMCEQCM) ,Q/2) for any M e M . If me a(M) then EQ(Rm) = 0 and so _M(Rra,W) = 0 . R— R— This means xTT(Rm) = Rm and so me xri(M) . On the other hand i f W W m e x (M) then ^(RnijW) = 0 and M(EQ(Rm) ,Q/Z) = 0 . By [16,p.89] W . K— A— we have EQ(Rm) = 0 . Consequently Rm is a-torsion. Hence a = x W and we get the desired similarity by Lemma (5.4) . Conversely, let U e F . Then R/U is x T-torsion which J -a W means „M(R/U,W) = 0 . But 0 = DM(R/U,W) - „M(EQ(R) @„ R/U,^/Z) K— K— £— R implies EQ(R) R/U = 0 . Hence by Theorem (4.5,vii) a has prope-rfey-T .. Tachikawa [30] mentions the following result: (5.6) Proposition: The localization of any M in M relative to x = x i s given by EQ(M) = { x e I(M/xM) | $ (x) = 0 for a l l <j) e $ } where $ = { (j) e M(I(M/ M) ,V) | f(M/xM) = 0 } and V is any injective similar to I'(S)' . Proof: . Putting C = M/xM , the Proposition follows from ED(C)/C = T v(I(C)/C) = f ) { ker(f) | f e M(I(C)/C,V) } and that TT : M(I(C)/C,V) >• $ , induced by the canonical projection TT : 1(C) > I(C)/C is a bijection since we can f i l l the diagram 46, 0 »• C y 1(C) > i ( c ) / C y 0 V with a unique g for any <j> e $ If V i s any injective in M , i t is tempting to try and construct injective resolutions from products of copies of V as far as possible because the further we can push such a resolution, the more closely the module being resolved i s pinned down by the. torsion theory associated with V . (5.7) Definition: Let U e M . We say that an R-module X has R— U-dominant dimension 2 n [23] ( notation: U-dom.dim(X) j> n ) i f there is an exact sequence 0 —y X — y X, — y ... —y X 1 n such that each X^ is a product of copies of U . (5.8) Lemma: If V and W are s i m i l a r i n j e c t i v e R-modules, then V-dom.dim(X) £ n if and only if W-dom.dim(X) £ n for every, X e M . Proof: Suppose V-dom.dim(X) > n .Then there is an exact sequence 0 — y X y X.. —y...—y X such that each X. is a product of copies I n I V V of V . Then each X_^  is cogenerated by W because of the similarity and the fact that a product of monos is mono in M . From R— [17,Lemma A.4,p.87] we conclude that W-dom.dim(X) > n . The coverse-implication follows by symmetry I Out of the similarity class of an injective, we wish to pick out a distinguished representative having the finiteness condition: 47. (5.9) Definition: An R-module X is called f i n i t e l y cogenerating [23] i f there is a f i n i t e number of elements f e M(R,X) i = l,...,n such that P\ ( ker(g) | g e M(R,X) } = f) { ker(f ±) | i = l,...,n } Notice that this is. the dual of f i n i t e l y generated. Since (o:x) = ker(g) for g e M(R,X) defined by r \ > rx x e X and M(R,X) - X by f i > f(1) , X is f i n i t e l y cogenerating i f and only i f there exists elements x,,...,x e X such that I n (5.10) (0:X) = Pl { (0:xi) | i = l,...,n } . (5.11) Lemma: Every i n j e c t i v e in M is similar to a finitely R— cogenerating i n j e c t i v e . V Proof: Suppose V is any injective R-module. Let W = V be the V-fold product of copies of V .. Obiously W and V are similar. If £ is the element of W whose v~" coordinate i s v for any v e V , then (0:W) = (0:5) and W is f i n i t e l y cogenerating. 1 Since the dominant dimension of a module is uniform over a similarity class of injectives in the sense of Lemma (5.8), we make the following: (5.12) Definition: For a s IKF(R) we say that an R-module X has q-dominant dimension £ n ( notation: c-dom.dim(X) £ n ) i f V-dom.dim(X) > n for any injective V such that a = . Using this terminology and keeping Lemma. (5.8) in mind, we state: (5.13) Proposition: For any a z IKF(R) and M e M i) M is o-torsion-free if and only if a-dom.dim(M) £ 1 i i ) M is faithfully a-injective if and only if a-dom.dim(M) £ 2 , 48. Proof is given in [17,Prop A.6,p.88] and [23,Lemmas 5.1 & 5.2] # This Proposition provides yet another characterization of the fa i t h f u l l y a-injective modules. Thus by Lemma (5.11) the f u l l subcategory. p_(V) of M consisting of modules with V-dom.dim > 2 for a f i n i t e l y cogenerating injective V considered by Morita [23] is exactly our category A^ associated with a = x^ . The fact that A is an abelian subcategory of M i f and only i f EQ is exact, a condition which does not hold in general, caused trouble in Section 3. Property T is a good attempt at patching up this d i f f i c u l t y and also provides some fringe benifits. In [23,Thm 6.1] Morita gives another answer to this problem: for a e IKF(R) , the category A is an abelian subcategory of M i f and only i f a-dom.dim(A) = °° for every A e A . Lemma (5.11) t e l l s us that for any idempotent kernel functor a there is sone f i n i t e l y cogenerating injective V such that a = x^ This method of picking a distinguished representative from the similarity class of an injective i s exploited in [23,Thm 5.6] as follows: (5.14) Theorem: ( Morita ) Let a e IKF(R) and V a finitely cogenerating injective such that a = x^ . I f DC i s the double centralizer of V , then EQ(R) = DC where EQ i s the localization functor relative to a . (5.15) Observation: From this Theorem and Proposition (3.12) we get very cheaply a result of Kato [15,Thm 2] which says that a fa i t h f u l f i n i t e l y cogenerating injective V has the double centralizer 49. property i f and only i f R is fai t h f u l l y x^-injective. However this i s a particular case of a more general result [23,Thm 3.4] for which we have to work considerably harder. The Theorem (5.14) no longer holds in general i f we drop the condition that V be f i n i t e l y cogenerating. However i f W is any injective such, that a =• x^ and DC the double centralizer of W , we do in fact obtain a ring homomorphism EQ(R) >• DC which is a monomorphism ! ( see 118] ) . (5.16) Example: For any ring R , the injective hull I(R) is always f i n i t e l y cogenerating injective. Then Theorem (5.14) says that EQ(R) relative to T„ is the double centralizer i f I(R) . This i s exactly the definition of the complete ring of quotients given-- ±ar [46 vpv94»]«v • We shall return to this Example again later, but f i r s t a generalization of the notion of dense l e f t ideal. (5.17) Definition: Let I be a l e f t ideal in R . A l e f t ideal J in R i s called I-dense i f V r ^ I Vr2 e R 3 r e R 3~ r r ^ i I and r r £ e J . (5.18) Proposition: ( Popescu ) For any left ideal I in R the strongly complete filter F_ associated with x^ for N = R/l consists exactly of the I-dense left ideals. Proof is given in [26] of this Proposition and of the following: (5.19) Corollary: If A i s a commutative ring, I an ideal, N = A/l and F the strongly complete filter associated with x^ then U e F if and only if a e A such that UaC I implies a e I . 50. (5.20) Example: Recall that in the Goldie Torsion Theory the R-module N = R/z2(R) has zero singular ideal and so is Z^-torsion-f -tree. This means 2^ ^  TN ' ^ n o r < ^ e r t 0 P r o v e that the converse implication also holds, suppose U is a Z^-dense l e f t ideal in R , and l e t r I U If r e Z2(R) , then Er <k Z^ (R) for some essential E If Er = 0 then (U:r) 3 (0:r) D E is essential. If Er ^  0 then 0 •/= xr z (R) for some x e E which we can pick so that xr i U ( otherwise (U:r) 2 E is essential already ). Now there is an essential E' such that E'xr = 0 e U so (U:xr) 3 E' is also essential. Suppose r i Z^(R) . If (U:r) is not essential, there i s a l e f t ideal B + 0 such that (U:r) D B = 0 . Let 0 ^  b e B , so br i U . If br t Z 2(R) then because U is Z2(R)-dense there exists - x* such^tliat" • xbr-^s""Z-^('R)r-and' xbT~>e'~U«--- lev xb-"-/-'0' , x-b e^ -B" and xb e (U:r) - impossible. Hence br e Z^(R) . This means Ebr C Z^(R) for some essential E , which i s what we want , for i f Ebr = 0 then (U:br) £ (0:br) P E is essential already. If Ebr ^  0 then 0 7^  xbr e Z^(R) for some x e E and we can pick x such that xbr i U (otherwise (U:br) IP E is essential ). Now there is an essential E' such that E'xbr =0 so (U:xbr) P E' is essential. In any case, we can always find s e R such that (U:sr) is proper essential . Hence by Lemma (1.15,ii) every Z2(R)-dense l e f t ideal i s maxi i e . x^ ^  Z 2 . This shows = Z^ is the largest torsion theory for which R/Z2(R) is torsion-free, and we obtain another characterization of the maxi l e f t ideals as being Z2(R)-dense . (5.21) Example: The 0-dense l e f t ideals are just the usual dense l e f t ideals. Hence the strongly complete f i l t e r of dense l e f t ideals i s 51. exactly the f i l t e r associated with T r .For the case T = T r , comparing Proposition (5.6) with [16,Prop l,p.94] i t is clear that EQ(R) is the complete ring of l e f t quotients, which we denote here by Q^ . This agrees with the conclusion of Example (5.16). Since R is t R-torsion-free, we consider R as a subring of . Let a e IKF(R) such that a(R) =0 . The resulting localised ring EQ(R) relative to such a a is called a f a i t h f u l ( l e f t ) quotient ring of R . The importance of the complete ring of quotients of R comes from the fact that i t exists for every ring R; ( which was seen not to be the case for the classical ring of quotients ) and that any fa i t h f u l quotient ring of R is a subring of Q^ .This is because o(R) = 0 implies a < T D SO that every l e f t ideal in F R ' — must be dense. Now By Proposition (3"".TO')' EQ(R) = { x e I(R) | (R:x) e F^ } Q { x e I(R) | (R:x) is dense} = (5.22) Proposition: The idempotent kernel functor T R has property T if and only if Q^  has no proper dense left i d e a l s . Proof: Suppose T d has property T . Let D be a dense l e f t ideal in Q^  . Then R O D is dense in R . To see this, take r^ ^ 0 and r in R . Now there exists q e Q such that qr.. ^ 0 and qr. e D . m 1 2' For some dense U in R , Uq R by Proposition (3.10) and Uqr^ =/ 0 as Q^ is torsion-free. Hence for some, u e U , uqr^ f 0 and uqr^ e R D which shows that R O D is dense. Now Q = EQ(RH D) = Q 0 _ (ROD) m TU K = (^(RH D) as Q m i s f l a t in ^ C QJD C D and so Q^ has no proper dense l e f t ideals. 52. Conversely, l e t D be dense in R . Then Q D is dense in m : take q^ f 0 , in . For some dense J in R we have Jq^ C R and J q 2 R with Jq^ f 0 . So 0 f aq^ and aq^ are in R for some a e J . As D is dense, there, exists r e R such that raq^ f 0 and. raq^ e D C Q^ D . This shows Q^ D is dense in and so Q^ D = . By Theorem (4.5,iii) we conclude that x^ has property T . I (5.23) Remark: The above Proposition has an obvious generalization: i f 0 e IKF (R) with <j> : R > EQ(R) canonical then 0 has property T i f and only i f EQ(R) has no proper l e f t ideals J such that <j> "''(J) e F . The same argument as i n the Proposition shows this statement is equivalent to Theorem (4.5,iii) . One of the objectives of such a localization process is of course to determine "local properties". By this we mean the following: let SUB(R) be a subset of IKE(R) and suppose £2 is a property satisfied by the ring R ( or by an R-module M ). Then tt_ is called a SUB(R)-local property when R ( M ) has ft" i f and, only i f EQ.(R). ( EQ„(M)_ ). has fi, for every localiz.ation relative to a member of SUB(R) . Now the advantage of having every idempotent kernel functor in the form x allows us to distinguish a subset of IKF(R) by means of a distinguished class of modules, li k e simples or indecomposable injectives for example. This is done in [9] and in [26] by an equivalent method. Another objective of this localization is to obtain information about the structure of R via the structure of i t s localizations EQ(R) by imposing conditions on the rings R and EQ(R) - see for example [24]. A classical example of this i s of course the Goldie Theory ( as in [11,Chap 7] ). In this thesis the machinery of localization has been developed as i t is being used to date, along with some fundamental results and of course examples where, as usual, the real action of the theory is taking place. Indications are that the generalised concept of torsion with i t s relative localization w i l l prove i t s e l f increasingly valuable in the further study of rings and modules. 54. BIBLIOGRAPHY 1. M. F. ATIYAH and I. G. MacDONALD, "Introduction to Commutative Algebra", Addison-Westley, Reading, Mass., 1969. 2. N. BOURBAKI, "Elements de Mathematique, Fac XXVII", Hermann, Paris, 1961. 3. K. L. CHEW, Extensions of Rings and Modules, Thesis, University of British Columbia, 1965. 4. P. M. COHN, Rings of Fractions, Amer. Math. Monthly 78 (1971), 596-615. 5. S. E. DICKSON, A Torsion Theory for Abelian Categories, Trans. Amer. Math. Soc. 21 (1966), 223-235. 6. V. DLAB, The Concept of a Torsion Module, Amer. Math. Monthly 75 (1968), 973-976. 7. P. GABRIEL, Des Categories Abeliennes, Bull. Soc. Math. France 90 (1962), 326-448. 8. A. W. GOLDIE, Torsion-free Modules and Rings, J. Algebra 1 (1964) 268-287. 9. 0. GOLDMAN, Rings and Modules of Quotients, J. Algebra 13 (1969), 10-47. 10. A. G. HEINICKE, Triples and Localization, to appear 11. I. N. HERSTEIN, "Noncommutative Rings", Carus Monograph 15, 1968. 12. P. J. HILTON and U. STAMMBACH, "A Course in Homological Algebra", Springer-Verlag, New York, 1971. 13. J". P. JANS, "Rings and' Homology"", Holt RInehart and Winston, New York, 1964. 14. , Sone Aspects of Torsion, Pac. J. Math. 15 (1965), 1249-1259. 15. T. KATO, Rings of U-dominant dimension Z 1, Tohoku Math J. 21 (1969), 321-327. 16. J. LAMBEK, "Lectures on Rings and Modules", Gin(Blaisdelle), Waltham, Mass., 1966. 17. , Torsion Theories, Additive Semantics, and Rings of Quotients, Lecture Notes in Mathematics 177, Springer-Verlag, New York, 1971. 55. 18. , Bicommutators of Nice Injectives, J. Algebra 21 (1972), 60-73. 19. Noncommutative Localization, Alg. Symposium, Carleton University, 1972 (unpublished). 20. J. K. LUDEMAN, A Generalization of the Concept of a Ring of Quotients, Can. Math. Bull. 14 (1971), 517-529. 21. J.-M. MARANDA, Injective Structures, Trans. Amer. Math. Soc. 110 (1964), 98-135. 22. B. MITCHELL, "Theory of Categories", Academic Press, New York, 1965. 23. K. MORITA, Localizations in Categories of Modules I, Math Z. 114 (1970) , 121-144. 24. S. PAGE, Properties of Quotient Rings, to appear. 25. B. PAREIGIS, "Categories and Functors", Academic Press, New York, 1970. 26. N. POPESCU, Le Spectre a Gauche d'un Anneaux, J. Algebra 18 (1971) , 213-228-. 27. and T. SPIRCU, Epimorphisma D Anneaux, J. Algebra 16 (1970), 40-59. 28. L. SILVER, Noncommutative Localization and Applications, J. Algebra 7 (1967), 44-76. 29. B. STENSTROM, Rings and Modules of Quotients, Lecture Notes in Mathematics 237, Springer-Verlag, New York, 1971. •30. H. TACHIKAWA, Double Centralizers and Dominant Dimensions, Math. Z. 116 (1970), 79-88. 31. M. L. TEPLY, Some Aspects of Goldie's Torsion Theory, Pac. J. Math. ". -29 (1969), 447-459. 32. C. T. TSAI, Report on Injective Modules, Queen's Papers No. 6, Queen's University, 1965. 33. C. L. WALKER and E. WALKER, Quotient Categories and Rings of Quotients, to appear. 

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