Special Precovered Categories of Gorenstein Categories

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1 Special Precovered Categories of Gorenstein Categories Tiwei Zhao and Zhaoyong Huang Department of Mathematics, Nanjing University, Nanjing 9, Jiangsu Province, P. R. China Astract Let A e an aelian category and P(A ) the sucategory of A consisting of projective ojects. Let C e a full, additive and self-orthogonal sucategory of A with P(A ) a generator, and let G(C ) e the Gorenstein sucategory of A. Then the right -orthogonal category G(C ) of G(C ) is oth projectively resolving and injectively coresolving in A. We also get that the sucategory SPC(G(C )) of A consisting of ojects admitting special G(C )-precovers is closed under extensions and C -stale direct summands (*). Furthermore, if C is a generator for G(C ), then we have that SPC(G(C )) is the minimal sucategory of A containing G(C ) G(C ) with respect to the property (*), and that SPC(G(C )) is C -resolving in A with a C -proper generator C. Introduction As a generalization of finitely generated projective modules, Auslander and Bridger introduced in [] the notion of finitely generated modules of Gorenstein dimension zero over commutative Noetherian rings. Then Enochs and Jenda generalized it in [7] to aritrary modules over a general ring and introduced the notion of Gorenstein projective modules and its dual (that is, the notion of Gorenstein injective modules). Let A e an aelian category and C an additive and full sucategory of A. Recently Sather-Wagstaff, Sharif and White introduced in [4] the notion of the Gorenstein sucategory G(C ) of A, which is a common generalization of the notions of modules of Gorenstein dimension zero [], Gorenstein projective modules, Gorenstein injective modules [7], V -Gorenstein projective modules and V -Gorenstein injective modules [9], and so on. Let R e an associative ring with identity, and let Mod R e the category of left R-modules and G(P(Mod R) the sucategory of Mod R consisting of Gorenstein projective modules. Let PC(G(P(Mod R)) and SPC(G(P(Mod R)) e the sucategories of Mod R consisting of modules admitting a G(P(Mod R))- precover and admitting a special G(P(Mod R))-precover respectively. The following question in relative homological algera remains still open: does PC(G(P(Mod R)) = Mod R always hold true? Several authors have gave some partially positive answers to this question, see [, 4, 5, 6]. Note that in these references, PC(G(P(Mod R)) = SPC(G(P(Mod R)), see Example 4.8 elow for details. In particular, any module in Mod R with finite Gorenstein projective dimension admits a G(P(Mod R))-precover which is Mathematics Suject Classification: 8G5, 8E. Keywords: Gorenstein categories, Right -orthogonal categories, Special precovers, Special precovered categories, Projectively resolving, Injectively coresolving. address: tiweizhao@hotmail.com address: huangzy@nju.edu.cn

2 T. W. Zhao and Z. Y. Huang also special ([]). In fact, it is unknown whether PC(G(P(Mod R)) = SPC(G(P(Mod R)) always holds true. Based on the aove, it is necessary to study the properties of these two sucategories. Let A e an aelian category and C an additive and full sucategory of A. We use SPC(G(C )) to denote the sucategory of A consisting of ojects admitting special G(C )-precovers. The aim of this paper is to investigate the structure of SPC(G(C )) in terms of the properties of the right -orthogonal category G(C ) of G(C ). This paper is organized as follows. In Section, we give some terminology and some preliminary results. Assume that C is self-orthogonal and the sucategory of A consisting of projective ojects is a generator for C. In Section, we prove that G(C ) is oth projectively resolving and injectively coresolving in A. We also characterize when all ojects in A are in G(C ). In Section 4, we prove that SPC(G(C )) is closed under extensions and C -stale direct summands (*). Furthermore, if C is a generator for G(C ), then we get the following two results: () SPC(G(C )) is the minimal sucategory of A containing G(C ) G(C ) with respect to the property (*); and () SPC(G(C )) is C -resolving in A with a C -proper generator C. Preliminaries Throughout this paper, A is an aelian category and all sucategories of A are full, additive and closed under isomorphisms. We use P(A ) (resp. I (A )) to denote the sucategory of A consisting of projective (resp. injective) ojects. For a sucategory C of A and an oject A in A, the C -dimension C - dim A of A is defined as inf{n there exists an exact sequence C n C C A in A with all C i in C }. Set C - dim A = if no such integer exists (cf. []). For a non-negative integer n, we use C n (resp. C < ) to denote the sucategory of A consisting of ojects with C -dimension at most n (resp. finite C -dimension). Let X e a sucategory of A. Recall that a sequence in A is called Hom A (X, )-exact if it is exact after applying the functor Hom A (X, ) for any X X. Dually, the notion of a Hom A (, X )-exact sequence is defined. Set X := {M Ext A (X, M) = for any X X }, X := {M Ext A (M, X) = for any X X }, and X := {M Ext A (X, M) = for any X X }, X := {M Ext A (M, X) = for any X X }. We call X (resp. X ) the right (resp. left) -orthogonal category of X. Let X and Y e sucategories of A. We write X Y if Ext A (X, Y ) = for any X X and Y Y. Definition.. (cf. [6]) Let X Y e sucategories of A. The morphism f : X Y in A with X X and Y Y is called an X -precover of Y if Hom A (X, f) is epic for any X X. An X - precover f : X Y is called special if f is epic and Ker f X. X is called special precovering in Y

3 Special Precovered Categories of Gorenstein Categories if any oject in Y admits a special X -precover. Dually, the notions of a (special) X -(pre)envelope and a special preenveloping sucategory are defined. Definition.. (cf. []) A sucategory of A is called projectively resolving if it contains P(A ) and is closed under extensions and under kernels of epimorphisms. Dually, the notion of injectively coresolving sucategories is defined. From now on, assume that C is a given sucategory of A. Definition.. (cf. [4]) The Gorenstein sucategory G(C ) of A is defined as G(C ) = {M is an oject in A there exists an exact sequence: C C C C (.) in C, which is oth Hom A (C, )-exact and Hom A (, C )-exact, such that M = Im(C C )}; in this case, (.) is called a complete C -resolution of M. In what follows, R is an associative ring with identity, Mod R is the category of left R-modules and mod R is the category of finitely generated left R-modules. Remark.4. () Let R e a left and right Noetherian ring. Then G(P(mod R)) coincides with the sucategory of mod R consisting of modules with Gorenstein dimension zero ([]). () G(P(Mod R)) (resp. G(I (Mod R))) coincides with the sucategory of Mod R consisting of Gorenstein projective (resp. injective) modules ([7]). () Let R e a left Noetherian ring, S a right Noetherian ring and R V S a dualizing imodule. Put W = {V S P P P(Mod S)} and U = {Hom S(V, E) E I (Mod S op )}. Then G(W ) (resp. G(U )) coincides with the sucategory of Mod R consisting of V -Gorenstein projective (resp. injective) modules ([9]). Definition.5. (cf. [4]) Let X T e sucategories of A. Then X is called a generator (resp. cogenerator) for T if for any T T, there exists an exact sequence T X T (resp. T X T ) in T with X X ; and X is called a projective generator (resp. an injective cogenerator) for T if X is a generator (resp. cogenerator) for T and X T (resp. T X ). We have the following easy oservation. Lemma.6. Assume that C C and P(A ) is a generator for C. Then for any G G(C ), there exists a Hom A (C, )-exact and Hom A (, C )-exact exact sequence G P G in A with P P(A ) and G G(C ). Proof. Let G G(C ). Then there exists a Hom A (C, )-exact and Hom A (, C )-exact exact sequence G C G

4 4 T. W. Zhao and Z. Y. Huang in A with C C and G G(C ). Because P(A ) is a generator for C y assumption, there exists an exact sequence C P C in A with P P(A ) and C C. Consider the following pullack diagram C C G P G G C. G By [, Lemma.5], the middle row is oth Hom A (C, )-exact and Hom A (, C )-exact, and hence G G(C ) y [, Proposition 4.7], that is, the middle row is the desired sequence. The following result is useful in the sequel. Proposition.7. Assume that C C and P(A ) is a generator for C. Then () G(C ) = G(C ). () G(C ) C C. Proof. () It suffices to prove that G(C ) G(C ). Let M G(C ) and G G(C ). By Lemma.6, we have an exact sequence G P G in A with P P(A ) and G G(C ). It induces Ext A (G, M) = Ext A (G, M) =, and hence Ext A (G, M) = and Ext A (G, M) = Ext A (G, M) =. Repeating this process, we get Ext A (G, M) =. () See [, Lemma 5.7]. We remark that if A has enough projective ojects, and if P(A ) C and C is closed under kernels of epimorphisms, then P(A ) is a generator for C. The right -orthogonal category of G(C ) In the rest of this paper, assume that the sucategory C is self-orthogonal (that is, C C ) and P(A ) is a generator for C. In this section, we mainly investigate the homological properties of G(C ). We egin with some examples of G(C ). Example.. () By Proposition.7 and [, Theorem 5.8], we have P(A ) C C < G(C ). () P(A ) < I (A ) < G(C ).

5 Special Precovered Categories of Gorenstein Categories 5 () If the gloal dimension of R is finite, then G(P(Mod R)) = Mod R. (4) By [8, Theorem.5.] and [, Theorem.9], we have that R is quasi-froenius if and only if G(P(Mod R)) = I (Mod R), and if and only if G(P(Mod R)) = P(Mod R) = I (Mod R). For a non-negative integer n, recall that a left and right noetherian ring R is called n-gorenstein if the left and right self-injective dimensions of R are at most n. The following result is a generalization of Example.(4). Example.. If R is n-gorenstein, then G(P(Mod R)) = P(Mod R) n = P(Mod R) < = I (Mod R) n = I (Mod R) <. Proof. By [, Theorem ] and Example.(), we have P(Mod R) n = P(Mod R) < = I (Mod R) n = I (Mod R) < G(P(Mod R)). Now let M G(P(Mod R)) and N Mod R. Since R is n-gorenstein, there exists an exact sequence G n P n P M in Mod R with all P i in P(Mod R) and G n G(P(Mod R)) y [8, Theorem.5.]. Then we have Ext n+ R (N, M) = Ext R(G n, M) = and M I (Mod R) n, and thus G(P(Mod R)) I (Mod R) n. The following result shows that G(C ) ehaves well. Theorem.. () G(C ) is closed under direct products, direct summands and extensions. () G(C ) is projectively resolving in A. () G(C ) is injectively coresolving in A. Proof. () It is trivial. () By Example.(), P(A ) G(C ). Let G G(C ) and L M N e an exact sequence in A with M, N G(C ). By Proposition.7(), we have Ext A (G, M) = = Ext A (G, N). Then Ext A (G, L) =. Because G G(C ), we have an exact sequence G C G in A with C C and G G(C ). For C, there exists an exact sequence C P C

6 6 T. W. Zhao and Z. Y. Huang in A with P P(A ) and C C. Consider the following pullack diagram C C G P G G C. G By the aove argument, we have Ext A (G, L) = Ext A (G, L) =. Because the leftmost column splits y Proposition.7(), G is isomorphic to a direct summand of G and Ext A (G, L) =, which shows that L G(C ). () It is trivial that I (A ) G(C ). By Proposition.7, we have that G(C ) is closed under cokernels of monomorphisms. Thus G(C ) is injectively coresolving. Before giving some applications of Theorem.(), consider the following example. Example.4. Let Q e a quiver: a a a and I =< a a a, a a a >. Let R = kq/i with k a field. Then the Auslander-Reiten quiver Γ(mod R) of mod R is as follows. Γ(mod R) : By a direct computation, we have G(P(mod R)) : , G(P(mod R)) : , where the terms marked y circles are indecomposale projective modules in mod R. G(P(mod R)) G(P(mod R)) = P(mod R). Then we have

7 Special Precovered Categories of Gorenstein Categories 7 In general, we have the following Corollary.5. If C is closed under direct summands, then for any n, we have G(C ) n G(C ) = C n. Proof. By Example.(), we have C n G(C ) n G(C ). Now let M G(C ) n G(C ). By [, Theorem 5.8], there exists an exact sequence K n C n C M in A with all C i in C and K n G(C ). By Theorem.(), we have K n G(C ). Because C is closed under direct summands y assumption, it follows easily from the definition of G(C ) that K n C and M C n. Proposition.6. For any M A, the following statements are equivalent. () M G(C ). () The functor Hom A (, M) is exact with respect to any short exact sequence in A ending with an oject in G(C ). () Every short exact sequence starting with M is Hom A (G(C ), )-exact. If, moreover, R is a commutative ring, A = Mod R and C = P(Mod R), then the aove conditions are equivalent to the following (4) Hom R (Q, M) G(P(Mod R)) for any Q P(Mod R). Proof. () () () It is easy. Now let R e a commutative ring. () (4) For any G G(P(Mod R)), we have an exact sequence in Mod R with P P(Mod R). Let Q P(Mod R). Then K f P G (.) Q R K Q f Q R P Q R G is exact. It is easy to check that Q R G G(P(Mod R)). Then Ext R(Q R G, M) = y (), and so Hom R ( Q f, M) is epic. By the adjoint isomorphism, we have that Hom R (f, Hom R (Q, M)) is also epic. So applying the functor Hom R (, Hom R (Q, M)) to (.) we get Ext R(G, Hom R (Q, M)) =, and hence Hom R (Q, M) G(P(Mod R)). (4) () It is trivial y setting Q = R. In the following result, we characterize categories over which all ojects are in G(C ). Proposition.7. Assume that C is closed under direct summands. Consider the following conditions. () G(C ) = A. () G(C ) G(C ). () G(C ) = C. Then we have () () (). If C is a projective generator for A, then all of them are equivalent.

8 8 T. W. Zhao and Z. Y. Huang Proof. The implication () () is trivial. () () Let G G(C ). Then there exists an exact sequence G C G in A with C C and G G(C ). By (), we have that G G(C ) and the aove exact sequence splits. Thus as a direct summand of C, G C y assumption. If C is a projective generator for A, then the implication () () follows directly. Let X e a sucategory of mod R containing P(mod R). We use X to denote the stale category of X modulo P(mod R). We end this section y giving two examples aout G(P(mod R)). Example.8. Let Q and Q e the following two quivers Q : α α α 4 Q : a α a α c α c d d, α α and let I =< α α, α α, α 4 α, α4 > and I =< α α a, α a α >. Let R = KQ /I and R = KQ /I. Note that R is Gorenstein and R is not Gorenstein. The Auslander-Reiten quivers of mod R and mod R are as follows. Γ(mod R ) :, Γ(mod R ) : d c c a c a d a c c a d c a c a d c a d. a Then we have () The ojects marked in a cycle or a ox are indecomposale ojects in G(P(mod R i )) (i =, ); in particular, the ojects marked in a cycle are indecomposale ojects in P(mod R i ) (i =, ). mod R () modr modr and G(P(mod R )) mod R G(P(mod R )). () G(P(mod R )) G(P(mod R )) and G(P(mod R )) G(P(mod R )).

9 Special Precovered Categories of Gorenstein Categories 9 Example.9. Let Q and Q e the following two quivers Q : α α α 4 α 4 α a α Q : a c, α d α c d and let I =< α α, α 4 α, α α 4 > and I =< α c α, α d α c, α α d >. Let R = KQ /I and R = KQ /I. Then the Auslander-Reiten quivers of mod R and mod R are as follows. Γ(mod R ) : , Γ(mod R ) : a a c c c a d 4444 a d d c d c, Then we have () The ojects marked in a cycle or a ox are indecomposale ojects in G(P(mod R i )) (i =, ); in particular, the ojects marked in a cycle are indecomposale ojects in P(mod R i ) (i =, ). mod R () modr modr and G(P(mod R )) mod R G(P(mod R )). () G(P(mod R )) G(P(mod R )) and G(P(mod R )) G(P(mod R )). 4 The special precovered category of G(C ) In this section, we introduce and investigate the special precovered category of G(C ) in terms of the properties of G(C ). Proposition 4.. () Let M G(C ) and f : C M e an epimorphism in A with C C. Then Ker f G(C ) and f is a special G(C )-precover of M. () Consider an exact sequence M C M. (4.) If M admits special G(C )-precover, then so is M. The converse is true if C is a generator for G(C ) and (4.) is Hom A (C, )-exact. Proof. () The assertion follows from Example.() and Theorem.(). () Assume that M admits a special G(C )-precover and N G M

10 T. W. Zhao and Z. Y. Huang is an exact sequence in A with G G(C ) and N G(C ). Comining it with the following Hom A (, C )- exact exact sequence G i C p G in A with C C and G G(C ), we get the following commutative diagram with exact columns and rows N G M. i C p G g h C M Adding the exact sequence C C C to the middle row, we otain the following commutative diagram with exact columns and rows N G ( i ) C C M C, ( p (g, C ) C ) G C h M which can e completed to a commutative diagram with exact columns and rows as follows. N G M C ( i ) C C ( p C ) G C M C (g, C ) M h. Note that G C G(C ). Moreover, since N G(C ), we have M G(C ) y Theorem.(). Thus the rightmost column in the aove diagram is a special G(C )-precover of M.

11 Special Precovered Categories of Gorenstein Categories Now let C e a generator for G(C ) special G(C )-precover and and (4.) e Hom A (C, )-exact. Assume that M admits a L G M, L C L are exact sequences in A with G G(C ), L G(C ) following commutative diagram with exact columns and rows L G M C C C C L G M. and C C. By [, Lemma.()], we get the By Proposition.7() and Theorem.(), we have L G(C ) and the leftmost column is Hom A (C, )- exact. So the middle column is also Hom A (C, )-exact. On the other hand, the middle column is Hom A (, C )-exact y Proposition.7(). So G G(C ) y [, Proposition 4.7(5)], and hence the upper row is a special G(C )-precover of M. We introduce the following Definition 4.. We call SPC(G(C )) := {A A precovered category of G(C ). A admits a special G(C )-precover} the special It is trivial that SPC(G(C )) is the largest sucategory of A such that G(C ) is special precovering in it. In particular, SPC(G(C )) = A if and only if G(C ) is special precovering in A. For the sake of convenience, we say that a sucategory X of A is closed under C -stale direct summands provided that the condition X C X with C C implies X X. Theorem 4.. () SPC(G(C )) is closed under extensions. () SPC(G(C )) is closed under C -stale direct summands. Proof. () Let L M N e an exact sequence in A. Assume that L and N admit special G(C )-precovers and L G L f L, N G N g N

12 T. W. Zhao and Z. Y. Huang are exact sequences in A with G L, G N G(C ) and L, N G(C ). Consider the following pullack diagram L Q G N α L M N. g Since Ext R(G N, L ) = y Proposition.7(), we get an epimorphism Ext R(G N, f) : Ext R(G N, G L ) Ext R(G N, L). It induces the following commutative diagram with exact rows G L G M G N f β L Q G N L M N. α g Set M := Ker αβ. Then we get the following commutative diagram with exact columns and rows L M N G L G M G N L M N. Note that G M G(C ) (y [4, Corollary 4.5]) and M G(C ) (y Theorem.()). Thus the middle column in the aove diagram is a special G(C )-precover of M. This proves that SPC(G(C )) is closed under extensions. () Let M SPC(G(C )) and K G M e an exact sequence in A with G G(C ) and K G(C ). Assume that M = L C with C C, we have an exact and split sequence C M L

13 Special Precovered Categories of Gorenstein Categories in A. Consider the following pullack diagram K L C K G M L L. Since K, C G(C ), we have L G(C ) diagram is a special G(C )-precover of L. y Theorem.(). Thus the middle column in the aove The following question seems to e interesting. Question 4.4. Is SPC(G(C )) closed under direct summands? The following result shows that SPC(G(C )) possesses certain minimality, which generalizes [5, Theorem 6.8()]. Theorem 4.5. Assume that C is a generator for G(C ). Then we have () G(C ) G(C ) SPC(G(C )) and SPC(G(C )) is closed under extensions and C -stale direct summands. () SPC(G(C )) is the minimal sucategory with respect to the property () as aove. To prove this theorem, we need the following Lemma 4.6. Let K G M e an exact sequence in A with K G(C ) and G G(C ). Then there exists an exact sequence G M C K in A with K G(C ) and C C. Proof. Let K G M e an exact sequence in A with K G(C ) and G G(C ). Since G G(C ), there exists a Hom A (C, )- exact exact sequence G C G

14 4 T. W. Zhao and Z. Y. Huang in A with C C and G G(C ). Consider the following pushout diagram K G K C G M K G. Since K, C G(C ), we have K G(C ) y Theorem.(). Consider the following pullack diagram M G G Q C M K G. Since the middle column in the first diagram is Hom A (C, )-exact, so is the rightmost column in this diagram. Then the middle row in the second diagram is also Hom A (C, )-exact y [, Lemma.4()], and in particular, it splits. Thus Q = M C and the middle column in the second diagram is the desired exact sequence. Proof of Theorem 4.5. () It follows from Proposition 4.() and Theorem 4.. () Let X e a sucategory of A such that G(C ) G(C ) X and X is closed under extensions and C -stale direct summands. Let M SPC(G(C )). Then y Lemma 4.6, we have an exact sequence G M C K in A with K G(C ), G G(C ) and C C. Because G, K X, we have that M C X and M X. It follows that SPC(G(C )) X. As an immediate consequence of Theorem 4.5, we get the following Corollary 4.7. Assume that G(P(Mod R)) is special precovering in Mod R and X is a sucategory of Mod R. If G(P(Mod R)) G(P(Mod R)) X and X is closed under extensions and P(Mod R)- stale direct summands, then X = Mod R.

15 Special Precovered Categories of Gorenstein Categories 5 Proof. By assumption, we have SPC(G(P(Mod R))) = Mod R. Now the assertion follows from Theorem 4.5. We collect some known classes of rings R satisfying that G(P(Mod R)) is special precovering in Mod R as follows. Example 4.8. For any one of the following rings R, G(P(Mod R)) is special precovering in Mod R. () Commutative Noetherian rings of finite Krull dimension ([5, Remark 5.8]). () Rings in which all projective left R-modules have finite injective dimension ([6, Corollary 4.]); especially, Gorenstein rings (that is, n-gorenstein rings for some n ). () Right coherent rings in which all flat R-modules have finite projective dimension ([, Theorem.5] and [4, Proposition 8.]); especially, right coherent and left perfect rings, and right Artinian rings. We recall the following definition from []. Definition 4.9. Let C, T and E e sucategories of A with C T. () C is called an E -proper generator (resp. E -coproper cogenerator) for T if for any oject T in T, there exists a Hom A (E, ) (resp. Hom A (, E ))-exact exact sequence T C T (resp. T C T ) in A such that C is an oject in C and T is an oject in T. () T is called E -preresolving in A if the following conditions are satisfied. (i) T admits an E -proper generator. (ii) T is closed under E -proper extensions, that is, for any Hom A (E, )-exact exact sequence A A A in A, if oth A and A are ojects in T, then A is also an oject in T. An E -preresolving sucategory T of A is called E -resolving if the following condition is satisfied. (iii) T is closed under kernels of E -proper epimorphisms, that is, for any Hom A (E, )-exact exact sequence A A A in A, if oth A and A are ojects in T, then A is also an oject in T. In the following, we investigate when SPC(G(C )) is C -resolving. We need the following two lemmas. Lemma 4.. For any M SPC(G(C )), there exists a Hom A (C, )-exact exact sequence K C M in A with C C. Proof. Let M SPC(G(C )). Then there exists a Hom A (C, )-exact exact sequence K G M in A with G G(C ) and K G(C ). For G, there exists a Hom A (C, )-exact exact sequence G C G

16 6 T. W. Zhao and Z. Y. Huang in A with C C and G G(C ). Consider the following pullack diagram G G K C M K G. M By [, Lemma.5], the middle row is Hom A (C, )-exact, as desired. Lemma 4.. Assume that C is a generator for G(C ). Given a Hom A (C, )-exact exact sequence in A, we have () If M, N SPC(G(C )), then L SPC(G(C )). L M N () If L, M SPC(G(C )) and there exists a Hom A (C, )-exact exact sequence in A with C C, then N SPC(G(C )). K C N Proof. Let L M N e a Hom A (C, )-exact exact sequence in A. () Assume that M, N SPC(G(C )). By Lemma 4., there exists a Hom A (C, )-exact exact sequence K C N in A with C C. Consider the following pullack diagram L K K T C L M N. By Proposition 4.(), K SPC(G(C )). Then it follows from Theorem 4.() and the exactness of the middle column that T SPC(G(C )). Notice that the middle row is Hom A (C, )-exact y [, Lemma.4()], so it splits and T = L C. Thus L SPC(G(C )) y Theorem 4.().

17 Special Precovered Categories of Gorenstein Categories 7 () Assume L, M SPC(G(C )) and there exists a Hom A (C, )-exact exact sequence K C N in A with C C. As in the aove diagram, since L, C SPC(G(C )), we have T SPC(G(C )) y Theorem 4.(). Moreover, the middle column is Hom A (C, )-exact y [, Lemma.4()]. So K SPC(G(C )) y (), and hence N SPC(G(C )) y Proposition 4.(). Now we are ready to prove the following Theorem 4.. If C is a generator for G(C ), then SPC(G(C )) is C -resolving in A with a C -proper generator C. Proof. Following Theorem 4.() and Lemma 4.(), we know that SPC(G(C )) is closed under C -proper extensions and kernels of C -proper epimorphisms. Now let M SPC(G(C )). Then y Lemma 4., there exists a Hom A (C, )-exact exact sequence K C M in A with C C. By Proposition 4.(), we have K SPC(G(C )). It follows that C is a C -proper generator for SPC(G(C )) and SPC(G(C )) is a C -resolving. As a consequence, we get the following Corollary 4.. If C is a projective generator for A, then SPC(G(C )) is projectively resolving and injectively coresolving in A. Proof. Let C e a projective generator for A. Because G(C ) is projectively resolving y Theorem.(), C is also a projective generator for G(C ). It follows from Theorem 4. that SPC(G(C )) is projectively resolving. Now let I e an injective oject in A and K P f I an exact sequence in A with P C. Then it is easy to see that K G(C ) y Example.() and Theorem.(). So f is a special G(C )-precover of I and I SPC(G(C )). On the other hand, y Lemma 4.(), we have that SPC(G(C )) is closed under cokernels of monomorphisms. Thus we conclude that SPC(G(C )) is injectively coresolving. The following corollary is an immediate consequence of Corollary 4., in which the second assertion generalizes [5, Theorem 6.8()]. Corollary 4.4. () SPC(G(P(Mod R))) is projectively resolving and injectively coresolving in Mod R. () If R is a left Noetherian ring, then SPC(G(P(mod R))) is projectively resolving and injectively coresolving in mod R.

18 8 T. W. Zhao and Z. Y. Huang Let SPE(G(C )) e the sucategory of A consisting of ojects admitting special G(C )-preenvelopes. We point out that the dual versions on G(C ) and SPE(G(C )) of all of the aove results also hold true y using completely dual arguments. Acknowledgements. This research was partially supported y NSFC (Grant No. 5764), a Project Funded y the Priority Academic Program Development of Jiangsu Higher Education Institutions, the University Postgraduate Research and Innovation Project of Jiangsu Province 6 (No. KYZZ6 4), Nanjing University Innovation and Creative Program for PhD candidate (No. 6). The authors thank the referees for the useful suggestions. References [] F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, Secondnd edition, Grad. Texts in Math., Springer-Verlag, Berlin, 99. [] J. Asadollahi, T. Dehghanpour and P. Hafezi, On the existence of Gorenstein projective precovers, Rend. Sem. Mat. Univ. Padova 6 (6), [] M. Auslander and M. Bridger, Stale module theory, Memoirs Amer. Math. Soc. 94, Amer. Math. Soc., Providence, RI, 969. [4] D. Bravo, J. Gillespie and M. Hovey, The stale module category of a general ring, Preprint is availale at arxiv: [5] L. W. Christensen, H.-B. Foxy and H. Holm, Beyond totally reflexive modules and ack: A survey on Gorenstein dimensions, Commutative algera Noetherian and Non-Noetherian Perspectives, Springer, New York,, pp. 4. [6] E. E. Enochs, Injective and flat covers, envelopes and resolvents, Israel J. Math. 9 (98), [7] E. E. Enochs and O. M. G. Jenda, Gorenstein injective and projective modules, Math. Z. (995), 6 6. [8] E. E. Enochs and O. M. G. Jenda: Relative Homological Algera, de Gruyter Exp. Math., Walter de Gruyter, Berlin, New York,. [9] E. E. Enochs, O. M. G. Jenda and J. A. López-Ramos, Covers and envelopes y V -Gorenstein modules, Comm. Algera (5), [] H. Holm, Gorenstein homological dimensions, J. Pure Appl. Algera 89 (4), [] Z. Y. Huang, Proper resolutions and Gorenstein categories, J. Algera 9 (), [] Z. Y. Huang, Homological dimensions relative to preresolving sucategories, Kyoto J. Math. 54 (4), [] Y. Iwanaga, On rings with finite self-injective dimension II, Tsukua J. Math. 4 (98), 7.

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