Ding Projective and Ding Injective Modules

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Ding Projective and Ding Injective Modules∗

Gang Yang School of Mathematics, Physics and Software Engineering Lanzhou Jiaotong University, Lanzhou 730070, China E-mail: [email protected] Zhongkui Liu College of Mathematics and Information Science Northwest Normal University, Lanzhou 730070, China E-mail: [email protected] Li Liang School of Mathematics, Physics and Software Engineering, Lanzhou Jiaotong University, Lanzhou 730070, China E-mail: [email protected]

Received 21 April 2010 Revised 13 September 2010 Communicated by Nanqing Ding

Abstract. An R-module M is called Ding projective if there exists an exact sequence 0 1 0 1 · · · → P1 → P0 → P → P → · · · of projective R-modules with M = Ker(P → P ) such that Hom(−,F ) leaves the sequence exact whenever F is a ﬂat R-module. In this paper, we develop some basic properties of such modules. Also, properties of Ding injective modules are discussed. 2010 Mathematics Subject Classiﬁcation: primary 16D40, 16D50; secondary 16E05, 16E65 Keywords: FP-injective modules, Ding-Chen rings, Ding projective modules, Ding injective modules

1 Introduction and Preliminaries Throughout the paper, we assume all rings have an identity and all modules are unitary. Unless stated otherwise, an R-module will be understood to be a left R-module. Let R be a ring. Recall from [7] that an R-module M is Gorenstein projective 0 1 if there exists an exact sequence · · · → P1 → P0 → P → P → · · · of projective

∗This work was partly supported by the NSF (11101197, 11261050, 11226059, 11201376) of China and the NSF of Gansu (1107RJZA233) of China. 602 G. Yang, Z.K. Liu, L. Liang

R-modules with M = Ker(P 0 → P 1) such that Hom(−,P ) leaves the sequence exact whenever P is a projective R-module. An R-module N is called Gorenstein 0 1 injective if there exists an exact sequence · · · → I1 → I0 → I → I → · · · of injective R-modules with N = Ker(I0 → I1) such that Hom(I, −) leaves the sequence exact whenever I is an injective R-module. There is a variety of nice results about Gorenstein projective and Gorenstein injective modules (see [6, 8, 9, 13]). Recently, Ding et al. [4, 5] considered two special cases of the Gorenstein projective and Gorenstein injective modules, which they called strongly Gorenstein flat and Gorenstein FP-injective modules, respectively. These two classes of modules over coherent rings possess many nice properties analogous to Gorenstein projective and Gorenstein injective modules over Noetherian rings. For example, when R is an n-Gorenstein ring (that is, R is a left and right Noetherian ring with self injective dimension at most n on both sides), Hovey [14] showed that the category of R- modules has two Quillen equivalent model structures, a projective model structure and an injective model structure. Similar results were shown by Gillespie in [11] when R is an n-FC ring (that is, R is a left and right coherent ring with self FP- injective dimension at most n on both sides). Because n-FC rings were introduced and studied by Ding and Chen in [2, 3] and seen to have many properties similar to n-Gorenstein rings, Gillespie called such rings Ding-Chen. Also, for the reason that Ding and co-authors introduced the notions of strongly Gorenstein flat and Gorenstein FP-injective modules, Gillespie renamed strongly Gorenstein flat as Ding projective, and Gorenstein FP-injective as Ding injective (see [11] for details). In this paper, we continue to study the properties of Ding projective and Ding injective modules. The paper is organized as follows: In Section 2, we introduce Ding projective and Ding injective R-modules, and use techniques different from those in [13] to show that over an arbitrary associate ring R, the class of Ding projective modules is projectively resolving, and if 0 → M → N → L → 0 is a short exact sequence of modules with M and N Ding projective, then L is Ding projective if and only if Ext1(L, F ) = 0 for all flat modules F . In Section 3, we are inspired by the facts that the projectivity and injectivity of modules can be characterized by applying commutative diagrams, to consider the similar characterizations of Ding projective and Ding injective modules. We first show that if R is a Ding-Chen ring, then an R-module M is Ding projective if and only if for any R-module X and any Ding injective preenvelope of X, f : X → B, any homomorphism α : M → C = Coker(f) can be lifted to β : M → B, i.e., we have the following completed commutative diagram:

M β α f ~ 0 / X / B / C / 0

Dually, properties concerning the Ding injective modules are investigated. In the end of this section, we show that if R is Ding-Chen, then every submodule of a Ding projective module is Ding projective if and only if every quotient of a Ding injective module is Ding injective. Ding Projective and Ding Injective Modules 603

We recall some notions and terminologies needed in the sequel. In [18], Stenströmintroduced the notion of an FP-injective module and studied FP-injective modules over coherent rings. An R-module E is called FP-injective if Ext1(A, E) = 0 for all finitely presented R-modules A. More generally, the FP- injective dimension of an R-module B is defined to be the least integer n ≥ 0 such that Extn+1(A, B) = 0 for all finitely presented R-modules A. The FP-injective dimension of B is denoted by FP-id(B) and equals ∞ if no such n above exists. FC rings, as the coherent version of quasi-Frobenius rings where Noetherian is replaced by coherent and self injective is replaced by self FP-injective, were introduced by Damiano in [1]. Just as Gorenstein rings are natural generalizations of quasi- Frobenius rings, Ding and Chen extended FC rings to n-FC rings in [2, 3] which are renamed as Ding-Chen rings by Gillespie [11].

Deﬁnition 1.1. A ring R is called an n-FC ring if it is both left and right coherent and FP-id(RR) and FP-id(RR) are both less than or equal to n. A ring R is called Ding-Chen if it is an n-FC ring for some n ≥ 0.

Examples of Ding-Chen rings include all Gorenstein rings and all von Neumann regular rings. In particular, if R is an infinite product of fields, then R is a Ding- Chen ring. Furthermore, it follows from [12, Theorem 7.3.1] that R[x1, x2, . . . , xn] is a commutative Ding-Chen ring. Another example of a Ding-Chen ring is the group ring R[G], where R is an FC-ring (i.e., 0-FC ring) and G is a locally finite group (see [1]). Given a class H of R-modules, we will denote the class of R-modules X satisfying Ext1(H,X) = 0 (respectively, Ext1(X,H) = 0) for every H ∈ H by H⊥ (respectively, ⊥H). Following [8], we give the following definitions:

Deﬁnition 1.2. A pair of classes of R-modules (A, B) is said to be a cotorsion pair if A⊥ = B and ⊥B = A.

Deﬁnition 1.3. A cotorsion pair (A, B) is said to be complete if for any R-module X, there are exact sequences 0 → X → B → A → 0 and 0 → Be → Ae → X → 0 with B, Be ∈ B and A, Ae ∈ A.

Deﬁnition 1.4. Let H be a class of R-modules and X an R-module. A homomorphism f : X → H is called an H-preenvelope if H ∈ H and the abelian group homomorphism Hom(f, H0) : Hom(H,H0) → Hom(X,H0) is surjective for each H0 ∈ H. An H-preenvelope f : X → H is called special if Ext1(Coker(f),H0) = 0 for all H0 ∈ H. Dually, H-precovers and special H-precovers can be deﬁned.

2 Ding Modules over General Rings Deﬁnition 2.1. An R-module M is called Ding projective if there exists an exact 0 1 sequence of projective R-modules P =: · · · → P1 → P0 → P → P → · · · with M = Ker(P 0 → P 1), which remains exact after applying Hom(−,F ) for any ﬂat R-module F . In this case, we say that P is a strongly complete projective resolution of M. We denote the class of all Ding projective R-modules by DP. 604 G. Yang, Z.K. Liu, L. Liang

Deﬁnition 2.2. An R-module N is called Ding injective if there exists an exact 0 1 sequence of injective R-modules I =: · · · → I1 → I0 → I → I → · · · with N = Ker(I0 → I1), which remains exact after applying Hom(E, −) for any FP- injective R-module E. In this case, we say that I is a strongly complete injective resolution of N. We denote the class of all Ding injective R-modules by DI.

Let R be a ring. For convenience, we will write “modules” to mean “left R- modules” in the rest of this paper unless otherwise specified. By definitions, Ding projective modules are Gorenstein projective and Ding injective modules are Gorenstein injective. If R is left Noetherian, then any FP- injective module is injective by [17, Theorem 1.6], and so any Gorenstein injective module is Ding injective. For a left coherent ring R, it follows from [8, Proposition 10.2.6] that a finitely presented module is Ding projective if and only if it is Goren- stein projective. Clearly, any Gorenstein projective module over a left perfect ring is Ding projective. Also, it follows easily from [11, Corollary 4.6] that any Gorenstein projective module over a Gorenstein ring is Ding projective.

Deﬁnition 2.3. [13] We call a class of modules X projectively resolving if all projective modules are contained in X , and for every short exact sequence 0 → X0 → X → X00 → 0 with X00 ∈ X , the conditions X ∈ X and X0 ∈ X are equivalent. We call a class of modules Y injectively resolving if all injective modules are contained in Y, and for every short exact sequence 0 → Y 0 → Y → Y 00 → 0 with Y 0 ∈ Y, the conditions Y 00 ∈ Y and Y ∈ Y are equivalent.

It was shown in [13] that the class of Gorenstein projective (respectively, Goren- stein injective) modules is projectively (respectively, injectively) resolving. One can prove that the similar results hold for Ding projective and Ding injective modules by using the proofs of [13, Theorems 2.5] and its dual version [13, Theorems 2.6]. In the following, we will give new proofs of these facts.

Lemma 2.4. Let 0 → M 0 → M → M 00 → 0 be an exact sequence of modules. If M 0 and M 00 are Ding projective, then so is M. Proof. The proof is similar to that of [6, Lemma 3.1]. ¤

Lemma 2.5. A module M is Ding projective if and only if there exists an exact sequence of modules 0 → M → P → N → 0 such that P is projective and N is Ding projective. Proof. The necessity is clear. To see the suﬃciency, let

ν 0 / M / P / N / 0 (\)

be an exact sequence of modules with P projective and N Ding projective. Then for any ﬂat module F , we get Exti(N,F ) = 0 for all i ≥ 1 by [4, Lemma 2.4], and hence we have Exti(M,F ) = 0 for all i ≥ 1 by using the long exact sequence

Exti(P,F ) / Exti(M,F ) / Exti+1(N,F ) . Ding Projective and Ding Injective Modules 605

Since N is Ding projective, there exists an exact sequence of projective modules

µ 0 / N / P 1 / P 2 / ··· , ([) which remains exact after applying Hom(−,F ) for any ﬂat module F . Assembling the sequences (\) and ([), we get an exact sequence of projective modules

µν 0 / M / P / P 1 / P 2 / ··· , (†) which remains exact after applying Hom(−,F ) for any flat module F . On the other hand, since Exti(M,F ) = 0 for all flat modules F and all i ≥ 1, the projective resolution of M / / / / ··· P1 P0 M 0 (‡) remains exact after applying Hom(−,F ) for any flat module F . Assembling the sequences (†) and (‡), we get an exact sequence of projective modules

µν / / / / 1 / 2 / ··· P1 P0 P P P ··· with M = Ker(µν), which remains exact after applying Hom(−,F ) for any ﬂat module F . So M is Ding projective. ¤

Theorem 2.6. The class DP of Ding projective modules is projectively resolving. Proof. Clearly, every projective module is Ding projective. Now consider a short exact sequence of modules 0 → M 0 → M → M 00 → 0 with M 00 ∈ DP. If M 0 ∈ DP, then by Lemma 2.4 we have M ∈ DP. If M ∈ DP, then by Lemma 2.5, there exists a short exact sequence of modules 0 → M → P → N → 0 with P projective and N ∈ DP. Consider the following pushout diagram:

0 0

0 / M 0 / M / M 00 / 0

0 / M 0 / P / A / 0

N N

0 0 Since the class DP is closed under extensions by Lemma 2.4, we get from the right vertical sequence that A is Ding projective. Therefore, by the middle horizontal sequence and Lemma 2.5, M 0 is Ding projective. This proves that the class DP is projectively resolving. ¤ 606 G. Yang, Z.K. Liu, L. Liang

Corollary 2.7. The class DP is closed under direct summands. Proof. Since the class DP is easily seen to be closed under arbitrary direct sums by the deﬁnition, and is projectively resolving, the result then follows from [13, Proposition 1.4]. ¤

Dual arguments to the above give the following assertions concerning the class DI of Ding injective modules.

Theorem 2.8. The class DI is injectively resolving.

Corollary 2.9. The class DI is closed under direct summands.

Holm proved in [13] that if 0 → M → N → L → 0 is a short exact sequence of modules with M and N Gorenstein projective, then L is Gorenstein projective if and only if Ext1(L, P ) = 0 for all projective modules P . One can prove that the similar result holds for Ding projective modules by using methods in [13] which involve homological dimensions. Here, we will give a new proof to it.

Theorem 2.10. Let 0 → M → N → L → 0 be a short exact sequence of modules. If M and N are Ding projective, then L is Ding projective if and only if Ext1(L, F ) = 0 for all flat modules F . Proof. The necessity follows from [4, Lemma 2.4]. We now prove the sufficiency. Let f g 0 / M / N / L / 0 be a short exact sequence of modules with M and N Ding projective, and Ext1(L, F ) = 0 for all flat modules F . Then there exist exact sequences of modules

M =: 0 / M / P 0 / P 1 / ··· , N =: 0 / N / Q0 / Q1 / ···

with each P i and Qi projective, which remain exact after applying Hom(−,F ) for any ﬂat module F . It is easy to see that the homomorphism f : M → N can be lifted to a chain map α : M → N . We let C denote the mapping cone of α : M → N . Since α : M → N is a quasi-isomorphism (both M and N are exact), the long exact sequence of homology for the mapping cone shows that C is exact. Also, the sequence C of modules remains exact after applying Hom(−,F ) for any ﬂat module F since both M and N are so. Consider the following commutative diagram:

D =: 0 / M M / 0 / ···

C =: 0 / M / N ⊕ P 0 / Q0 ⊕ P 1 / ···

L =: 0 / 0 / K / Q0 ⊕ P 1 / ··· Ding Projective and Ding Injective Modules 607

where K = Coker(M → N ⊕ P 0). Clearly, the sequence

0 / D / C / L / 0

is exact. Since the sequences D and C are exact, we get that L is exact. Moreover, L remains exact after applying Hom(−,F ) for any ﬂat module F since both D and C are so. It is easy to see that there exists a homomorphism h : K → L such that the following diagram commutes:

0 / M / N ⊕ P 0 / K / 0

π h f g 0 / M / N / L / 0

where π : N ⊕ P 0 → N is the canonical projection. By the ﬁve lemma, we get that h is epimorphic. By the snake lemma, we have Ker(h) =∼ Ker(π) = P 0. Thus, the sequence of modules

h 0 / P 0 / K / L / 0

is exact. Moreover, this short exact sequence splits by assumption Ext1(L, P 0) = 0. Hence, K =∼ L ⊕ P 0. On the other hand, one can check that Exti(L, F ) = 0 for all ﬂat modules F and all i ≥ 1, and so Exti(K,F ) =∼ Exti(L ⊕ P 0,F ) = 0 for all ﬂat modules F and all i ≥ 1, thus the projective resolution / / / / K =: ··· P1 P0 K 0

of K remains exact after applying Hom(−,F ) for any ﬂat module F . Assembling the sequences K and L, we get the strongly complete projective resolution of K, and so K =∼ L ⊕ P 0 is Ding projective. Then by Corollary 2.7, L is Ding projective, as desired. ¤

We also have a dual version of Theorem 2.10 as follows.

Theorem 2.11. Let 0 → M → N → L → 0 be a short exact sequence of modules. If N and L are Ding injective, then M is Ding injective if and only if Ext1(E,M) = 0 for all FP-injective modules E.

3 Ding Modules over Ding-Chen Rings It is well known that the projectivity and injectivity of modules can be characterized by applying commutative diagrams. In this section we give analogous characterizations for the Ding projectivity and Ding injectivity of modules over Ding-Chen rings. According to [2, Corollary 3.18], if R is both left and right coherent, and FP-id(RR) and FP-id(RR) are both ﬁnite, then FP-id(RR) = FP-id(RR). Ding and Chen went on to prove the following result (see also [11, Theorem 4.2]). 608 G. Yang, Z.K. Liu, L. Liang

Theorem 3.1. Let R be an n-FC ring and M a module. Then the following are equivalent: (1) fd(M) < ∞. (2) fd(M) ≤ n. (3) FP-id(M) < ∞. (4) FP-id(M) ≤ n.

By the above, we see that for a Ding-Chen ring R, the class of all modules with finite flat dimension and the class of all modules with finite FP-injective dimension are the same, and we use W to denote these classes. Also, it is easy to see that W is both projectively resolving and injectively resolving. Ding and Mao proved that (⊥W, W) forms a complete cotorsion pair when R is a Ding-Chen ring [15, Theorem 3.8]. Also, (W, W⊥) forms a complete cotorsion pair when R is a Ding- Chen ring [16, Theorem 3.4]. Moreover, Gillespie proved the following fundamental result which leads to the next corollary.

Lemma 3.2. [11, Corollaries 4.5 and 4.6] Let R be a Ding-Chen ring. Then the following statements hold: (1) A module M is Ding projective if and only if M ∈ ⊥W. (2) A module N is Ding injective if and only if N ∈ W⊥.

Corollary 3.3. Let R be a Ding-Chen ring. Then the following statements hold: (1) Any module X has a special Ding projective precover. (2) Any module Y has a special Ding injective preenvelope.

The next two lammas play an important role in proving our main results in this section.

Lemma 3.4. [10, Lemma 4.1] Suppose that we have a morphism of short exact sequences as shown:

α β 0 / A / B / C / 0

f g h µ ν 0 / X / Y / Z / 0

(1) If f is an isomorphism, then the right square is both a pullback and a pushout square. (2) If h is an isomorphism, then the left square is both a pullback and a pushout square.

We note that any Ding injective preenvelope is monomorphic since DI contains all the injective modules, and any Ding projective precover is epimorphic since DP contains all the projective modules.

Lemma 3.5. Let R be a Ding-Chen ring. Then the following assertions hold: Ding Projective and Ding Injective Modules 609

(1) If M is a Ding projective module, then for any module X and any special Ding injective preenvelope of X, f : X → B, any homomorphism α : M → C = Coker(f) can be lifted to β : M → B. That is, we have the following completed commutative diagram:

M β α f ~ 0 / X / B / C / 0

(2) If N is a Ding injective module, then for any module X and any special Ding projective precover of X, g : A → X, any homomorphism α : K = Ker(g) → N can be extended to β : A → N. That is, we have the following completed commutative diagram:

NO ` β α g 0 / K / A / X / 0

Proof. (1) Let X be a module and f : X → B a special Ding injective preen- f g velope. Then the sequence 0 / X / B / C / 0 is exact, where C = Coker(f) ∈ W and g : B → C is the natural epimorphism. Since there is a µ short exact sequence 0 / K / P / C / 0 with P projective, it is easy to see that there exist homomorphisms ω : P → B and K → X such that the following diagram commutes:

µ 0 / K / P / C / 0

ω f g 0 / X / B / C / 0

Note that C and P are contained in the class W, then we have K ∈ W since W is projectively resolving, and hence Ext1(M,K) = 0. Thus, for any homomorphism α : M → C, there exists a homomorphism ν : M → P such that α = µν. If we put β = ων, then α = gβ, and this completes the proof. (2) It is dual to the proof of (1). ¤

Now we are in a position to give main results in this section which provide new equivalent characterizations of Ding projective and Ding injective modules.

Theorem 3.6. Let R be a Ding-Chen ring and M a module. Then the following statements are equivalent: (1) M is Ding projective. 610 G. Yang, Z.K. Liu, L. Liang

(2) For any module X and any Ding injective preenvelope of X, f : X → B, any homomorphism α : M → C = Coker(f) can be lifted to β : M → B. That is, we have the following completed commutative diagram:

M β α f ~ 0 / X / B / C / 0

Proof. (1)⇒(2) Suppose that X is any module, f : X → B is a Ding injective preenvelope of X, and α : M → C = Coker(f) is a homomorphism. Let f 0 : X → B0 be a special Ding injective preenvelope of X, then we have the following completed commutative diagram:

f π 0 / X / B / C / 0

µ0 ν0 0 f π0 0 / X / B0 / C0 / 0 where C0 = Coker(f 0), π and π0 are natural epimorphisms. By Lemma 3.5, there exists a homomorphism γ : M → B0 such that ν0α = π0γ. It follows from Lemma 3.4 and the property of the pullback square that we have a homomorphism β : M → B satisfying α = πβ. (2)⇒(1) For any module W ∈ W, consider an extension of W by M:

µ ν 0 / W / X / M / 0 .

f g By Corollary 3.3, there is an exact sequence 0 / W / B / C / 0 of modules such that f : W → B is a special Ding injective preenvelope of W , that is, B is Ding injective and C is contained in W. Clearly, B is also contained in W because both C and W are so, and hence B is injective by [11, Proposition 3.4]. Thus, there is a homomorphism α : X → B such that f = αµ. By the factor lemma, we have a homomorphism β : M → C satisfying βν = gα, i.e., the following completed diagram commutes:

µ ν 0 / W / X / M / 0

α β f g 0 / W / B / C / 0 By the condition (2), we have a homomorphism γ : M → B with β = gγ. Thus, there is a homomorphism h : X → W such that hµ = 1W by [19, Lemma 7.16]. µ ν This means that the sequence 0 / W / X / M / 0 splits, and so Ext1(M,W ) = 0. Then by Lemma 3.2, M is Ding projective. ¤

Theorem 3.7. Let R be a Ding-Chen ring and N a module. Then the following statements are equivalent: Ding Projective and Ding Injective Modules 611

(1) N is Ding injective. (2) For any module X and any Ding projective precover of X, g : A → X, any homomorphism α : K = Ker(g) → N can be extended to β : A → N. That is, we have the following completed commutative diagram:

NO ` β α g 0 / K / A / X / 0

Proof. It is dual to the proof of Theorem 3.6. ¤

Recall that a ring R is called left hereditary if every left ideal is projective. It is well known that R is left hereditary if and only if every submodule of a projective module is projective, if and only if every quotient of an injective module is injective.

Theorem 3.8. Let R be a Ding-Chen ring. Then the following statements are equivalent: (1) Every submodule of a Ding projective module is Ding projective. (2) Every quotient of a Ding injective module is Ding injective. Proof. (1)⇒(2) Let N be a Ding injective module and L ≤ N. We will show that N/L is Ding injective. Let X be any module. Then by Corollary 3.3, there exists τ an exact sequence 0 / K / M / X / 0 such that M → X is a Ding projective precover and K ∈ W. On the other hand, (1) implies that K is Ding projective, hence it is projective by [11, Lemma 2.2]. If f : K → N/L is any homomorphism, then there is a homomorphism h : K → N such that f = πh, where π : N → N/L is the natural epimorphism. Since N is Ding injective, it follows from Lemma 3.5 that there is a homomorphism h0 : M → N satisfying h = h0τ. Now let g = πh0 : M → N/L and we have f = gτ. Assume that α : M 0 → X is any Ding projective precover of X, and let β : K0 = Ker(α) → N/L be a homomorphism. Then we have the following completed commutative diagram:

τ 0 / K / M / X / 0

µ ν

τ 0 α 0 / K0 / M 0 / X / 0

where τ 0 is the natural injection. By arguments above, there exists a homomorphism γ : M → N/L such that βµ = γτ. It follows from Lemma 3.4 and the property of the pushout square that we have a homomorphism η : M 0 → N/L satisfying β = ητ 0. Thus, it follows from Theorem 3.7 that N/L is Ding injective, as desired. (2)⇒(1) It is dual to the proof of (1)⇒(2). ¤

Acknowledgements. The authors would like to thank the referee and Professor Nanqing Ding for many constructive suggestions which have improved the present paper. 612 G. Yang, Z.K. Liu, L. Liang

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