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Indecomposable Decompositions of Finitely Presented Pure-Injective Modules
192, 200]208 Ž1997. JA966978
JOURNAL OF ALGEBRA ARTICLE NO.
Indecomposable Decompositions of Finitely Presented Pure-Injective Modules Jose Pardo ´ L. Gomez ´ Departamento de Alxebra, Uni¨ ersidade de Santiago, 15771 Santiago de Compostela, Spain
and Pedro A. Guil Asensio Departamento de Matematicas, Uni¨ ersidad de Murcia, 30100 Espinardo, Murcia, Spain ´ Communicated by Kent R. Fuller Received February 13, 1996
I. INTRODUCTION It is well known that pure-injective Žs algebraically compact. modules behave well in relation to direct sum decompositions. Any indecomposable pure-injective module has local endomorphism ring w14x and, more generally, any pure-injective module has the exchange property w15x. But, in contrast with S-pure-injective modules, which always have indecomposable decompositions w14x, pure-injective modules need not have this property. In fact, if all pure-injective right R-modules have such a decomposition, then R is a right pure-semisimple ring by w13x. Obviously, even a right pure-injective ring R need not have an indecomposable decomposition} think of the endomorphism ring of an infinite-dimensional vector space. It is clear from the preceding considerations that, given a right pure-injective ring R, such a decomposition exists for R R if and only if R is semiperfect. However, while a right S-pure-injective ring is semiprimary with maximum condition on annihilator right ideals, a right pure-injective ring is only Von Neumann regular modulo the radical with the idempotent-lifting property 200 0021-8693r97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.
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in general w14x. It is thus natural to ask for conditions for a right pure-injective ring R to be semiperfect. On the other hand, taking a clue from Osofsky’s theorem that shows that if every cyclic right R-module is injective then R is semisimple w10x, we study as in w5x completely pure-injective modules, i.e., modules M such that every pure quotient of M is pure-injective. Similarly, a ring R will be called right completely pure-injective when R R is completely pureinjective. The class of completely pure-injective modules falls between the classes of pure-injective and S-pure-injective modules. However, there are commutative completely pure-injective rings which are not perfect, e.g., the ring of p-adic integers. In fact, if R is semiperfect, then the pure quotients of R R are direct summands and so a semiperfect right pureinjective ring is completely pure-injective. The preceding considerations and Osofsky’s theorem thus lead to the following question: Are right completely pure-injective rings semiperfect? In this paper, we provide an affirmative answer to this question. In fact, the first of our problems is more intimately related to the second than is obvious at first sight: a right pure-injective ring is semiperfect precisely when it is right completely pure-injective. These conditions can be transferred, using functorial techniques, to the endomorphism ring of a finitely presented pure-injective module, given the following necessary and sufficient conditions for the existence of an indecomposable decomposition for these modules. They constitute our main result. Namely, a finitely presented pure-injective right R-module M has an indecomposable decomposition if and only if it is completely pure-injective; these conditions are in turn equivalent to each pure submodule of M being a direct summand of a direct sum of finitely presented modules. Thus we see that, although finitely presented completely pure-injective modules are rather far from being S-pure-injective, in general, they share with the latter modules the key property of being direct sums of modules with local endomorphism rings. In particular, the rings R such that every finitely presented right R-module is completely pure-injective are Krull]Schmidt rings in the sense of w8x. Our method of proof was suggested by Osofsky’s proof of her theorem in w10x. The situation is different, however. In fact, our proof gives, in the injective case, a different and probably simpler proof of Osofsky’s theorem. Throughout this paper, all rings R will be associative and with identity, and Mod-R will denote the category of right R-modules. By a module we will usually mean a right R-module and, whenever we want to emphasize the fact that M is a right R-module, we will write MR . We refer to w1x, w9x, and w11x, for all undefined notions used in the text.
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2. RESULTS We begin with an auxiliary lemma. We refer to w9x for the definition and basic properties of pure-injective envelopes. LEMMA 2.1. Let R be a right pure-injecti¨ e ring and Q, T right ideals of R such that Q l T s 0 and Q [ T is pure in R R . If PQ , PT are pure-injecti¨ e en¨ elopes of Q, T, respecti¨ ely, contained in R R , then PQ l PT s 0 and PQ [ PT is a direct summand of R R . Proof. Using functor ring techniques w6x Žsee also w11, 52.3x or w3, Lemma 2.1x., R can be represented as the endomorphism ring of an injective object E of a Grothendieck category C such that the following properties hold. The functor Hom C Ž E, y.: C ª Mod-R has a left adjoint ymR E: Mod-R ª C which takes pure exact sequences of Mod-R into Žpure. exact sequences of C . Moreover, a right R-module X is pureinjective if and only if X mR E is injective in C , and if PEŽ X . is a pure-injective envelope of X in Mod-R, then PEŽ X . mR E is an injective envelope of X mR E in C . Thus we see that if we identify R with End C Ž E . and PQ is a pure-injective envelope of Q contained in R, we have that PQ s Hom C Ž E, PQ m E ., where PQ m E is an injective envelope of Q mR E contained in E. Similarly, PT s Hom C Ž E, PT m E . with PT m E an injective envelope of T mR E contained in E. On the other hand, since Q [ T is pure in R R , we have that Q mR E and T mR E are subobjects of E such that Ž Q mR E . l ŽT mR E . s 0. Therefore, the injective envelopes of these subobjects have zero intersection, that is, Ž PQ mR E . l Ž PT mR E . s 0. From this it follows that PQ l PT s 0. Obviously, PQ [ PT is then a pure-injective envelope of the pure right ideal Q [ T and hence a direct summand of R R . THEOREM 2.2. Let R be a right pure-injecti¨ e ring. If R is not semiperfect, then there exist pure right ideals N, L of R and a homomorphism w : N ª RrL that cannot be extended to R. Proof. Let J s J Ž R . be the radical of R. Then RrJ is Von Neumann regular and idempotents lift modulo J w14, Theorem 9x. Furthermore, any countable set of orthogonal idempotents of RrJ can be lifted to an orthogonal set of idempotents of R Žsee, e.g., w1, Exercise 27.1x.. Thus our hypothesis implies that there exists an infinite set e i 4I of orthogonal idempotents in R. Now let L s [i g I Ž e i R .. Then, for each finite subset F : I, [F Ž e i R . s Ž S F e i . R is a direct summand of R R and so L is a pure right ideal. Next consider a nonempty subset A of I and let A9 s I y A. Since the idempotents e i 4i g A are orthogonal, it is clear that [A Ž e i R . and [A9 Ž e i R . are right ideals of R that satisfy the hypotheses of Lemma 2.1. Thus there
INDECOMPOSABLE DECOMPOSITIONS
exist pure-injective envelopes PA of right ideal T of R such that
203
[A Ž e i R . and PA9 of [A9 Ž e i R . and a
R R s PA [ PA9 [ T . Then there exists an idempotent element e A g R such that PA s e A R and PA9 [ T s Ž1 y e A . R. Furthermore, we have that for each i g A, e i g e A R and so e A e i s e i , while for i g A9, e i g Ž1 y e A . R and hence e A e i s 0. Let us now write I s D A g A A as an infinite union of infinite pairwise disjoint subsets. As in w10x we have that, by Zorn’s lemma, there exists a maximal subset K : 2 I with respect to the properties: 1. 2. 3.
A : K. < A < G / 0 for each A g K. < A l B < - / 0 for each A, B g K, A / B.
Let N be the right ideal of R defined by N s Ý A g K e A R. We claim that N is a pure right ideal. To see this consider A, B g K, A / B. Then A l B is finite, say A l B s i1 , . . . , i r 4 . Observe now that if C s B y i1 , . . . , i r 4 , then C l A s B and, as e B R is a pure-injective envelope of [B Ž e i R ., we may write e B R s e i1 R [ ??? [ e i rR [ X B , where X B is a pure-injective envelope of [C Ž e i R .. Since e A R is a pure-injective envelope of [A Ž e i R ., it follows from Lemma 2.1 that e A R l X B s 0 and that e A R q e B R s e A R q X B is a direct summand of R R . By induction we obtain that if A1 , . . . , A n g K, then e A 1 R q ??? qe A nR is a direct summand of R R . Thus N is a direct limit of these direct summands and hence a pure right ideal of R. Next, consider the quotient module NrL s Ý A g K ŽŽ e A R q L.rL.. We claim that this sum is direct. Suppose, then, that A, B1 , . . . , Bs g K are different. As we have just seen Ý sjs1 e B j R is a direct summand of R R , so that we may write Ý sjs1 e B j R s gR with g g R an idempotent. Furthermore, gR is a pure-injective envelope of [B Ž e i R . with B s D sjs1 Bj . To show that e A R l gR : L, observe that A l B s A l ŽD sjs1 Bj . s D sjs1Ž A l Bj . is finite set, say A l B s k 1 , . . . , k r 4 . Then we have as before that e B R s e k 1 R [ ??? [ e k rR [ X B , where e A R l X B s 0. Since e k 1 R [ ??? [ e k rR : e A R we obtain by modularity that e A R l gR s e A R l Ž e k 1 R [ ??? [ e k rR [ X B . s e k 1 R [ ??? [ e k rR : L. Thus the sum is indeed direct. Now, we finish our argument as in the proof of w10, Theoremx. We define a homomorphism c : NrL ª RrL by c Ž e A q L. s e A q L if A g
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A and c Ž e A q L. s L if A f A. Let p : N ª NrL be the canonical projection and let w s c (p : N ª RrL. Assume, by contradiction, that w has an extension w : R ª RrL, so that w ( u s w , where u: N ª R denotes the inclusion. Let then x g R such that w Ž1. s x q L. We have, for each A g K, xe A q L s w Ž e A . s w Ž e A . q
½
eA q L L
if A g A , if A f A .
Therefore we have that xe A s e A q l A with l A g L for A g A, and that xe A g L for A f A. Let now A g A. For each i g A, e A e i s e i and so e i xe i s e i xe A e i s e i Ž e A q l A . e i s e i e A e i q e i l A e i s e i q e i l A e i . Since l A g L, we obtain that, for almost all i g I, e i l A e i s 0. Thus, for almost all i g A, e i s e i xe i . Let A 0 s i g A < e i s e i xe i 4 , which is a cofinite subset of A and hence infinite. For each A g A, choose a element c A g A 0 and set C s c A < A g A 4 . Since A is infinite so is C and by the maximality of K there exists a set D g K such that D l C is infinite. It is clear that D f A and so xe D g L and hence e i xe D e i s 0 for almost all i g I. In particular, e i xe D e i s 0 for almost all i g D l C. But if i g D l C we have that because i g C, e i s e i xe i , and because i g D, e i xe i s e i xe D e i . Thus we obtain that e i s 0 for almost all i g D l C, which is a contradiction and proves and theorem. COROLLARY 2.3. Let R be a right pure-injecti¨ e ring. Then the following conditions are equi¨ alent: Ži. R is semiperfect. Žii. R is right completely pure-injecti¨ e. Žiii. E¨ ery pure right ideal of R is pure-projecti¨ e. Proof. If R is semiperfect, then by w11, 36.4 Ž1.x, every pure right ideal of R is a direct summand and so conditions Žii. and Žiii. hold. Conversely, it follows from Theorem 2.2 that Žii. implies Ži.. Finally, assume that Žiii. holds. Let N and L be pure right ideals of R, f : N ª RrL a homomorphism, and p: R ª RrL the canonical projection. Then our hypothesis implies that there exists g: N ª R such that f s p( g. Let u: N ª R be the inclusion. Since R R is pure-injective, there exists h: R ª R such that g s h( u. Then f s p( g s p( h( u and, since f has an extension to R we see that R is semiperfect by Theorem 2.2. Thus we have that Žiii. implies Ži. and this completes the proof. Remarks. Observe that, since a ring whose cyclic right modules are injective is Von Neumann regular, Osofsky’s theorem follows at once from Corollary 2.3.
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It is clear that a right pure-injective ring R need not be completely pure-injective: any nonsemisimple regular right self-injective ring provides an example. On the other hand, recall that a right R-module M is S-pure-injective when every direct sum of copies of M is pure-injective. Since the pure submodules of S-pure-injective modules are direct summands w9, Corollary 8.2x, it is clear that every S-pure-injective module is completely pure-injective. The converse, however, is far from being true. In fact, while a right pure-injective ring which is either right noetherian or semiperfect is completely pure-injective Žsince the pure right ideals are direct summands., a completely pure-injective ring need not be S-pureinjective even if it is commutative and either noetherian or semiprimary. An example of the first situation is provided by the power series ring R s K ww X 1 , . . . , X n xx over a field K. It is well known that R is linearly compact and hence pure-injective by w14, Proposition 1x. Since R is also noetherian, it is completely pure-injective as we have already remarked. However, R is not S-pure-injective, as it is not semiprimary Žsee w14x.. An example of the second situation was given in w12, Example 11x, where a commutative semiprimary pure-injective Žand hence completely pureinjective. ring R is constructed which is not S-pure-injective. LEMMA 2.4. Let M be a finitely presented right R-module and S s EndŽ MR .. Then the following assertions hold: Ži. If X is a set and p: M Ž X . ª Q a pure epimorphism, then the canonical homomorphism Hom R Ž M, Q . mS M ª Q is an isomorphism. Žii. If U is a flat right S-module, then the canonical homomorphism U ª Hom R Ž M, U mS M . is an isomorphism. Proof. Ži. By w11, 34.2 Ž2.x, Q is a direct limit of finite direct sums of copies of M. If F is a finite set, then the canonical morphism Hom R Ž M, M Ž F . . mS M ª M Ž F . is an isomorphism. Since M is finitely presented, the functor Hom R Ž M, y.: Mod-R ª Mod-S preserves direct limits and so Hom R Ž M, Q . mS M ª Q is also an isomorphism. The proof of Žii. is similar, using the fact that a flat module is a direct limit of finitely generated free modules. It follows from w4, Theorem 1x that a finitely presented completely pure-injective module with Von Neumann regular endomorphism ring has an indecomposable decomposition. In the following corollary we obtain necessary and sufficient conditions for the existence of such a decomposition without any hypothesis on the endomorphism ring. In particular we see that, although the class of completely pure-injective rings is much larger than that of S-pure-injective rings, finitely presented completely pure-injective modules share with S-pure-injective modules the good behavior vis-a-vis indecomposable decompositions. `
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COROLLARY 2.5. Let M be a finitely presented pure-injecti¨ e right Rmodule. Then the following conditions are equi¨ alent: Ži. M has an indecomposable decomposition. Žii. M is completely pure-injecti¨ e. Žiii. E¨ ery pure submodule of M is pure-projecti¨ e. Proof. Let S s EndŽ MR .. Since the endomorphism ring of an indecomposable pure-injective module is local w14, Theorem 9x, condition Ži. is equivalent to S being semiperfect by w1, 27.6x. In particular, if Ži. holds, every pure right ideal of S is a direct summand. Let now Q be a pure quotient of MR . By Lemma 2.4, the canonical morphism Hom R Ž M, Q . mS M ª Q is an isomorphism. Further, it is clear that Hom R Ž M, Q . is a pure quotient, and hence a direct summand, of SS . Thus we have that Q + Hom R Ž M, Q . mS M is a direct summand of MR and so it is clear that Ži. implies Žii. and Žiii.. Assume now that Žii. holds. If Z s SrX is a pure quotient of SS , then Z is flat and so, by Lemma 2.4, the canonical morphism a Z : Z ª Hom R Ž M, Z mS M . is an isomorphism. Then let q: U ª V be a pure monomorphism in Mod-S and let f : U ª Z be a homomorphism. It is clear that Z mS M is a pure quotient of M and hence, by hypothesis, pure-injective. Since q mS M: U mS M ª V mS M is pure in Mod-R, the homomorphism f mS M: U mS M ª Z mS M has an extension h: V mS M ª Z mS M, that is, f mS M s h(Ž q mS M .. By naturality, we have f s 1 1 Ž Ž . Ž ay f mS M .( a U s ay q ms Z (Hom R M, Z (Hom R M, h (Hom R M, y1 M .( a u s a Z (Hom R Ž M, h.( a V ( q, which shows that f extends to V and hence that Z is pure-injective. Thus S is right completely pure-injective and hence semiperfect by Corollary 2.3, so that Žii. implies Ži.. Finally, we show that Žiii. implies Ži.. Using Theorem 2.2 it suffices to show that if N and L are pure right ideals of S and f : N ª SrL is a homomorphism, then f can be lifted to S. Let p: S ª SrL be the canonical projection. Then N mS M is a pure submodule of M and hence pure-projective. Since p mS M is a pure epimorphism, we see that f mS M factors in the form f mS M s Ž p mS M .( h with h: N mS M ª M. Since M is finitely presented we have, as before, that a S and a S r L are Ž isomorphisms and hence that f s ay1 f mS M .( a N s S r L (Hom R M, 1 y1 Ž . Ž . Ž ay (Hom M, p m M (Hom M, h ( a s p( a Sr L R S R N S (Hom R M, h.( a N . Thus f has a lifting to S and hence S is semiperfect. In w8x, a ring R is called a Krull]Schmidt ring when every finitely presented right R-module is a direct sum of modules with local endomorphism rings Žthis condition if left-right symmetric.. It follows from Corol-
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lary 2.5 that any finitely presented completely pure-injective module has such a decomposition and so we obtain: COROLLARY 2.6. Let R be a ring such that e¨ ery finitely presented right R-module is completely pure-injecti¨ e. Then R is a Krull]Schmidt ring. Remarks. Among the rings that satisfy the hypothesis of the preceding corollary are both left pure-semisimple and right pure-semisimple rings Žsee w7, 8x for the left case., commutative S-pure-injective rings w14x, and commutative linearly compact rings. The latter have a Morita duality by w2x and using a standard argument Žas in w6, Exercise 7.10x. one can show that every reflexive Žand hence every finitely generated. module is pureinjective. These rings need not be perfect and so, in contrast with the other two classes of rings just mentioned, they need not be S-pure-injective. As a consequence of the Crawley]Jønsson]Warfield theorem w1, 26.5x, Krull] Schmidt rings have the additional property that every pure-projective module is Žuniquely by Azumaya’s theorem w1, 12.6x. a direct sum of finitely presented modules with local endomorphism rings. This was observed in w14x for commutative S-pure-injective rings.
ACKNOWLEDGMENTS The authors would like to thank Juan Martınez for his helpful comments. ´ Hernandez ´ This work was partially supported by the DGICYT ŽPB93-0515, Spain.. The first author was also partially supported by the European Community ŽContract CHRX-CT93-0091. and the Xunta de Galicia ŽXUGA 10502B94. and the second author by the C. A. de Murcia ŽPIB 94-25..
REFERENCES w1x F. W. Anderson and K. R. Fuller, ‘‘Rings and categories of modules,’’ 2nd ed., Springer-Verlag, New York, 1992. ´ Morita duality for commutative rings, Comm. Algebra 18 Ž1990., 1781]1788. w2x P. N. Anh, ´ and N. V. Dung, Some decomposition properties of injective and purew3x J. L. Garcia injective modules, Osaka J. Math. 31 Ž1994., 95]108. w4x J. L. Gomez Pardo, N. V. Dung, and R. Wisbauer, Complete pure injectivity and ´ endomorphism rings, Proc. Amer. Math. Soc. 118 Ž1993., 1029]1034. w5x J. L. Gomez Pardo and P. A. Guil Asensio, Endomorphism rings of completely ´ pure-injective modules, Proc. Amer. Math. Soc. 124 Ž1996., 2301]2309. w6x L. Gruson and C. U. Jensen, Dimensions cohomologiques reliees ´ aux foncteurs lim Ž i., ¤
in Lecture Notes in Mathematics, Vol. 867, pp. 234]294, Springer-Verlag, Berlin, 1981. w7x I. Herzog, Finitely presented right modules over a left pure-semisimple ring, Bull. London Math. Soc. 26 Ž1994., 333]338.
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w8x I. Herzog, A test for finite representation type, J. Pure Appl. Algebra 95 Ž1994., 151]182. w9x C. U. Jensen and H. Lenzing, ‘‘Model Theoretic Algebra,’’ Gordon and Breach, New York, 1989. w10x B. L. Osofsky, Noninjective cyclic modules, Proc. Amer. Math. Soc. 19 Ž1968., 1383]1384. w11x R. Wisbauer, ‘‘Grundlagen der Modul- und Ringtheorie,’’ Verlag Reinhard Fischer, Munchen, 1988. ¨ w12x W. Zimmermann, Ž S-. algebraic compactness of rings, J. Pure Appl. Algebra 23, Ž1982., 319]328. w13x B. Zimmermann-Huisgen, Rings whose right modules are direct sums of indecomposable modules, Proc. Amer. Math. Soc. 77 Ž1979., 191]197. w14x B. Zimmermann-Huisgen and W. Zimmermann, Algebraically compact rings and modules, Math. Z. 161 Ž1978., 81]93. w15x B. Zimmermann-Huisgen and W. Zimmermann, Classes of modules with the exchange property, J. Algebra 88 Ž1984., 416]434.
JOURNAL OF ALGEBRA ARTICLE NO.
Indecomposable Decompositions of Finitely Presented Pure-Injective Modules Jose Pardo ´ L. Gomez ´ Departamento de Alxebra, Uni¨ ersidade de Santiago, 15771 Santiago de Compostela, Spain
and Pedro A. Guil Asensio Departamento de Matematicas, Uni¨ ersidad de Murcia, 30100 Espinardo, Murcia, Spain ´ Communicated by Kent R. Fuller Received February 13, 1996
I. INTRODUCTION It is well known that pure-injective Žs algebraically compact. modules behave well in relation to direct sum decompositions. Any indecomposable pure-injective module has local endomorphism ring w14x and, more generally, any pure-injective module has the exchange property w15x. But, in contrast with S-pure-injective modules, which always have indecomposable decompositions w14x, pure-injective modules need not have this property. In fact, if all pure-injective right R-modules have such a decomposition, then R is a right pure-semisimple ring by w13x. Obviously, even a right pure-injective ring R need not have an indecomposable decomposition} think of the endomorphism ring of an infinite-dimensional vector space. It is clear from the preceding considerations that, given a right pure-injective ring R, such a decomposition exists for R R if and only if R is semiperfect. However, while a right S-pure-injective ring is semiprimary with maximum condition on annihilator right ideals, a right pure-injective ring is only Von Neumann regular modulo the radical with the idempotent-lifting property 200 0021-8693r97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.
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in general w14x. It is thus natural to ask for conditions for a right pure-injective ring R to be semiperfect. On the other hand, taking a clue from Osofsky’s theorem that shows that if every cyclic right R-module is injective then R is semisimple w10x, we study as in w5x completely pure-injective modules, i.e., modules M such that every pure quotient of M is pure-injective. Similarly, a ring R will be called right completely pure-injective when R R is completely pureinjective. The class of completely pure-injective modules falls between the classes of pure-injective and S-pure-injective modules. However, there are commutative completely pure-injective rings which are not perfect, e.g., the ring of p-adic integers. In fact, if R is semiperfect, then the pure quotients of R R are direct summands and so a semiperfect right pureinjective ring is completely pure-injective. The preceding considerations and Osofsky’s theorem thus lead to the following question: Are right completely pure-injective rings semiperfect? In this paper, we provide an affirmative answer to this question. In fact, the first of our problems is more intimately related to the second than is obvious at first sight: a right pure-injective ring is semiperfect precisely when it is right completely pure-injective. These conditions can be transferred, using functorial techniques, to the endomorphism ring of a finitely presented pure-injective module, given the following necessary and sufficient conditions for the existence of an indecomposable decomposition for these modules. They constitute our main result. Namely, a finitely presented pure-injective right R-module M has an indecomposable decomposition if and only if it is completely pure-injective; these conditions are in turn equivalent to each pure submodule of M being a direct summand of a direct sum of finitely presented modules. Thus we see that, although finitely presented completely pure-injective modules are rather far from being S-pure-injective, in general, they share with the latter modules the key property of being direct sums of modules with local endomorphism rings. In particular, the rings R such that every finitely presented right R-module is completely pure-injective are Krull]Schmidt rings in the sense of w8x. Our method of proof was suggested by Osofsky’s proof of her theorem in w10x. The situation is different, however. In fact, our proof gives, in the injective case, a different and probably simpler proof of Osofsky’s theorem. Throughout this paper, all rings R will be associative and with identity, and Mod-R will denote the category of right R-modules. By a module we will usually mean a right R-module and, whenever we want to emphasize the fact that M is a right R-module, we will write MR . We refer to w1x, w9x, and w11x, for all undefined notions used in the text.
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2. RESULTS We begin with an auxiliary lemma. We refer to w9x for the definition and basic properties of pure-injective envelopes. LEMMA 2.1. Let R be a right pure-injecti¨ e ring and Q, T right ideals of R such that Q l T s 0 and Q [ T is pure in R R . If PQ , PT are pure-injecti¨ e en¨ elopes of Q, T, respecti¨ ely, contained in R R , then PQ l PT s 0 and PQ [ PT is a direct summand of R R . Proof. Using functor ring techniques w6x Žsee also w11, 52.3x or w3, Lemma 2.1x., R can be represented as the endomorphism ring of an injective object E of a Grothendieck category C such that the following properties hold. The functor Hom C Ž E, y.: C ª Mod-R has a left adjoint ymR E: Mod-R ª C which takes pure exact sequences of Mod-R into Žpure. exact sequences of C . Moreover, a right R-module X is pureinjective if and only if X mR E is injective in C , and if PEŽ X . is a pure-injective envelope of X in Mod-R, then PEŽ X . mR E is an injective envelope of X mR E in C . Thus we see that if we identify R with End C Ž E . and PQ is a pure-injective envelope of Q contained in R, we have that PQ s Hom C Ž E, PQ m E ., where PQ m E is an injective envelope of Q mR E contained in E. Similarly, PT s Hom C Ž E, PT m E . with PT m E an injective envelope of T mR E contained in E. On the other hand, since Q [ T is pure in R R , we have that Q mR E and T mR E are subobjects of E such that Ž Q mR E . l ŽT mR E . s 0. Therefore, the injective envelopes of these subobjects have zero intersection, that is, Ž PQ mR E . l Ž PT mR E . s 0. From this it follows that PQ l PT s 0. Obviously, PQ [ PT is then a pure-injective envelope of the pure right ideal Q [ T and hence a direct summand of R R . THEOREM 2.2. Let R be a right pure-injecti¨ e ring. If R is not semiperfect, then there exist pure right ideals N, L of R and a homomorphism w : N ª RrL that cannot be extended to R. Proof. Let J s J Ž R . be the radical of R. Then RrJ is Von Neumann regular and idempotents lift modulo J w14, Theorem 9x. Furthermore, any countable set of orthogonal idempotents of RrJ can be lifted to an orthogonal set of idempotents of R Žsee, e.g., w1, Exercise 27.1x.. Thus our hypothesis implies that there exists an infinite set e i 4I of orthogonal idempotents in R. Now let L s [i g I Ž e i R .. Then, for each finite subset F : I, [F Ž e i R . s Ž S F e i . R is a direct summand of R R and so L is a pure right ideal. Next consider a nonempty subset A of I and let A9 s I y A. Since the idempotents e i 4i g A are orthogonal, it is clear that [A Ž e i R . and [A9 Ž e i R . are right ideals of R that satisfy the hypotheses of Lemma 2.1. Thus there
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exist pure-injective envelopes PA of right ideal T of R such that
203
[A Ž e i R . and PA9 of [A9 Ž e i R . and a
R R s PA [ PA9 [ T . Then there exists an idempotent element e A g R such that PA s e A R and PA9 [ T s Ž1 y e A . R. Furthermore, we have that for each i g A, e i g e A R and so e A e i s e i , while for i g A9, e i g Ž1 y e A . R and hence e A e i s 0. Let us now write I s D A g A A as an infinite union of infinite pairwise disjoint subsets. As in w10x we have that, by Zorn’s lemma, there exists a maximal subset K : 2 I with respect to the properties: 1. 2. 3.
A : K. < A < G / 0 for each A g K. < A l B < - / 0 for each A, B g K, A / B.
Let N be the right ideal of R defined by N s Ý A g K e A R. We claim that N is a pure right ideal. To see this consider A, B g K, A / B. Then A l B is finite, say A l B s i1 , . . . , i r 4 . Observe now that if C s B y i1 , . . . , i r 4 , then C l A s B and, as e B R is a pure-injective envelope of [B Ž e i R ., we may write e B R s e i1 R [ ??? [ e i rR [ X B , where X B is a pure-injective envelope of [C Ž e i R .. Since e A R is a pure-injective envelope of [A Ž e i R ., it follows from Lemma 2.1 that e A R l X B s 0 and that e A R q e B R s e A R q X B is a direct summand of R R . By induction we obtain that if A1 , . . . , A n g K, then e A 1 R q ??? qe A nR is a direct summand of R R . Thus N is a direct limit of these direct summands and hence a pure right ideal of R. Next, consider the quotient module NrL s Ý A g K ŽŽ e A R q L.rL.. We claim that this sum is direct. Suppose, then, that A, B1 , . . . , Bs g K are different. As we have just seen Ý sjs1 e B j R is a direct summand of R R , so that we may write Ý sjs1 e B j R s gR with g g R an idempotent. Furthermore, gR is a pure-injective envelope of [B Ž e i R . with B s D sjs1 Bj . To show that e A R l gR : L, observe that A l B s A l ŽD sjs1 Bj . s D sjs1Ž A l Bj . is finite set, say A l B s k 1 , . . . , k r 4 . Then we have as before that e B R s e k 1 R [ ??? [ e k rR [ X B , where e A R l X B s 0. Since e k 1 R [ ??? [ e k rR : e A R we obtain by modularity that e A R l gR s e A R l Ž e k 1 R [ ??? [ e k rR [ X B . s e k 1 R [ ??? [ e k rR : L. Thus the sum is indeed direct. Now, we finish our argument as in the proof of w10, Theoremx. We define a homomorphism c : NrL ª RrL by c Ž e A q L. s e A q L if A g
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A and c Ž e A q L. s L if A f A. Let p : N ª NrL be the canonical projection and let w s c (p : N ª RrL. Assume, by contradiction, that w has an extension w : R ª RrL, so that w ( u s w , where u: N ª R denotes the inclusion. Let then x g R such that w Ž1. s x q L. We have, for each A g K, xe A q L s w Ž e A . s w Ž e A . q
½
eA q L L
if A g A , if A f A .
Therefore we have that xe A s e A q l A with l A g L for A g A, and that xe A g L for A f A. Let now A g A. For each i g A, e A e i s e i and so e i xe i s e i xe A e i s e i Ž e A q l A . e i s e i e A e i q e i l A e i s e i q e i l A e i . Since l A g L, we obtain that, for almost all i g I, e i l A e i s 0. Thus, for almost all i g A, e i s e i xe i . Let A 0 s i g A < e i s e i xe i 4 , which is a cofinite subset of A and hence infinite. For each A g A, choose a element c A g A 0 and set C s c A < A g A 4 . Since A is infinite so is C and by the maximality of K there exists a set D g K such that D l C is infinite. It is clear that D f A and so xe D g L and hence e i xe D e i s 0 for almost all i g I. In particular, e i xe D e i s 0 for almost all i g D l C. But if i g D l C we have that because i g C, e i s e i xe i , and because i g D, e i xe i s e i xe D e i . Thus we obtain that e i s 0 for almost all i g D l C, which is a contradiction and proves and theorem. COROLLARY 2.3. Let R be a right pure-injecti¨ e ring. Then the following conditions are equi¨ alent: Ži. R is semiperfect. Žii. R is right completely pure-injecti¨ e. Žiii. E¨ ery pure right ideal of R is pure-projecti¨ e. Proof. If R is semiperfect, then by w11, 36.4 Ž1.x, every pure right ideal of R is a direct summand and so conditions Žii. and Žiii. hold. Conversely, it follows from Theorem 2.2 that Žii. implies Ži.. Finally, assume that Žiii. holds. Let N and L be pure right ideals of R, f : N ª RrL a homomorphism, and p: R ª RrL the canonical projection. Then our hypothesis implies that there exists g: N ª R such that f s p( g. Let u: N ª R be the inclusion. Since R R is pure-injective, there exists h: R ª R such that g s h( u. Then f s p( g s p( h( u and, since f has an extension to R we see that R is semiperfect by Theorem 2.2. Thus we have that Žiii. implies Ži. and this completes the proof. Remarks. Observe that, since a ring whose cyclic right modules are injective is Von Neumann regular, Osofsky’s theorem follows at once from Corollary 2.3.
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It is clear that a right pure-injective ring R need not be completely pure-injective: any nonsemisimple regular right self-injective ring provides an example. On the other hand, recall that a right R-module M is S-pure-injective when every direct sum of copies of M is pure-injective. Since the pure submodules of S-pure-injective modules are direct summands w9, Corollary 8.2x, it is clear that every S-pure-injective module is completely pure-injective. The converse, however, is far from being true. In fact, while a right pure-injective ring which is either right noetherian or semiperfect is completely pure-injective Žsince the pure right ideals are direct summands., a completely pure-injective ring need not be S-pureinjective even if it is commutative and either noetherian or semiprimary. An example of the first situation is provided by the power series ring R s K ww X 1 , . . . , X n xx over a field K. It is well known that R is linearly compact and hence pure-injective by w14, Proposition 1x. Since R is also noetherian, it is completely pure-injective as we have already remarked. However, R is not S-pure-injective, as it is not semiprimary Žsee w14x.. An example of the second situation was given in w12, Example 11x, where a commutative semiprimary pure-injective Žand hence completely pureinjective. ring R is constructed which is not S-pure-injective. LEMMA 2.4. Let M be a finitely presented right R-module and S s EndŽ MR .. Then the following assertions hold: Ži. If X is a set and p: M Ž X . ª Q a pure epimorphism, then the canonical homomorphism Hom R Ž M, Q . mS M ª Q is an isomorphism. Žii. If U is a flat right S-module, then the canonical homomorphism U ª Hom R Ž M, U mS M . is an isomorphism. Proof. Ži. By w11, 34.2 Ž2.x, Q is a direct limit of finite direct sums of copies of M. If F is a finite set, then the canonical morphism Hom R Ž M, M Ž F . . mS M ª M Ž F . is an isomorphism. Since M is finitely presented, the functor Hom R Ž M, y.: Mod-R ª Mod-S preserves direct limits and so Hom R Ž M, Q . mS M ª Q is also an isomorphism. The proof of Žii. is similar, using the fact that a flat module is a direct limit of finitely generated free modules. It follows from w4, Theorem 1x that a finitely presented completely pure-injective module with Von Neumann regular endomorphism ring has an indecomposable decomposition. In the following corollary we obtain necessary and sufficient conditions for the existence of such a decomposition without any hypothesis on the endomorphism ring. In particular we see that, although the class of completely pure-injective rings is much larger than that of S-pure-injective rings, finitely presented completely pure-injective modules share with S-pure-injective modules the good behavior vis-a-vis indecomposable decompositions. `
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COROLLARY 2.5. Let M be a finitely presented pure-injecti¨ e right Rmodule. Then the following conditions are equi¨ alent: Ži. M has an indecomposable decomposition. Žii. M is completely pure-injecti¨ e. Žiii. E¨ ery pure submodule of M is pure-projecti¨ e. Proof. Let S s EndŽ MR .. Since the endomorphism ring of an indecomposable pure-injective module is local w14, Theorem 9x, condition Ži. is equivalent to S being semiperfect by w1, 27.6x. In particular, if Ži. holds, every pure right ideal of S is a direct summand. Let now Q be a pure quotient of MR . By Lemma 2.4, the canonical morphism Hom R Ž M, Q . mS M ª Q is an isomorphism. Further, it is clear that Hom R Ž M, Q . is a pure quotient, and hence a direct summand, of SS . Thus we have that Q + Hom R Ž M, Q . mS M is a direct summand of MR and so it is clear that Ži. implies Žii. and Žiii.. Assume now that Žii. holds. If Z s SrX is a pure quotient of SS , then Z is flat and so, by Lemma 2.4, the canonical morphism a Z : Z ª Hom R Ž M, Z mS M . is an isomorphism. Then let q: U ª V be a pure monomorphism in Mod-S and let f : U ª Z be a homomorphism. It is clear that Z mS M is a pure quotient of M and hence, by hypothesis, pure-injective. Since q mS M: U mS M ª V mS M is pure in Mod-R, the homomorphism f mS M: U mS M ª Z mS M has an extension h: V mS M ª Z mS M, that is, f mS M s h(Ž q mS M .. By naturality, we have f s 1 1 Ž Ž . Ž ay f mS M .( a U s ay q ms Z (Hom R M, Z (Hom R M, h (Hom R M, y1 M .( a u s a Z (Hom R Ž M, h.( a V ( q, which shows that f extends to V and hence that Z is pure-injective. Thus S is right completely pure-injective and hence semiperfect by Corollary 2.3, so that Žii. implies Ži.. Finally, we show that Žiii. implies Ži.. Using Theorem 2.2 it suffices to show that if N and L are pure right ideals of S and f : N ª SrL is a homomorphism, then f can be lifted to S. Let p: S ª SrL be the canonical projection. Then N mS M is a pure submodule of M and hence pure-projective. Since p mS M is a pure epimorphism, we see that f mS M factors in the form f mS M s Ž p mS M .( h with h: N mS M ª M. Since M is finitely presented we have, as before, that a S and a S r L are Ž isomorphisms and hence that f s ay1 f mS M .( a N s S r L (Hom R M, 1 y1 Ž . Ž . Ž ay (Hom M, p m M (Hom M, h ( a s p( a Sr L R S R N S (Hom R M, h.( a N . Thus f has a lifting to S and hence S is semiperfect. In w8x, a ring R is called a Krull]Schmidt ring when every finitely presented right R-module is a direct sum of modules with local endomorphism rings Žthis condition if left-right symmetric.. It follows from Corol-
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lary 2.5 that any finitely presented completely pure-injective module has such a decomposition and so we obtain: COROLLARY 2.6. Let R be a ring such that e¨ ery finitely presented right R-module is completely pure-injecti¨ e. Then R is a Krull]Schmidt ring. Remarks. Among the rings that satisfy the hypothesis of the preceding corollary are both left pure-semisimple and right pure-semisimple rings Žsee w7, 8x for the left case., commutative S-pure-injective rings w14x, and commutative linearly compact rings. The latter have a Morita duality by w2x and using a standard argument Žas in w6, Exercise 7.10x. one can show that every reflexive Žand hence every finitely generated. module is pureinjective. These rings need not be perfect and so, in contrast with the other two classes of rings just mentioned, they need not be S-pure-injective. As a consequence of the Crawley]Jønsson]Warfield theorem w1, 26.5x, Krull] Schmidt rings have the additional property that every pure-projective module is Žuniquely by Azumaya’s theorem w1, 12.6x. a direct sum of finitely presented modules with local endomorphism rings. This was observed in w14x for commutative S-pure-injective rings.
ACKNOWLEDGMENTS The authors would like to thank Juan Martınez for his helpful comments. ´ Hernandez ´ This work was partially supported by the DGICYT ŽPB93-0515, Spain.. The first author was also partially supported by the European Community ŽContract CHRX-CT93-0091. and the Xunta de Galicia ŽXUGA 10502B94. and the second author by the C. A. de Murcia ŽPIB 94-25..
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