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Ding Projective and Ding Injective Modules See discussions, stats, and author profiles for this publication at: http://www.researchgate.net/publication/258373539 Ding Projective and Ding Injective Modules ARTICLE in ALGEBRA COLLOQUIUM · DECEMBER 2013 Impact Factor: 0.3 · DOI: 10.1142/S1005386713000576 CITATIONS READS 2 51 3 AUTHORS, INCLUDING: Gang Yang Li Liang Lanzhou Jiaotong University Lanzhou Jiaotong University 21 PUBLICATIONS 43 CITATIONS 18 PUBLICATIONS 25 CITATIONS SEE PROFILE SEE PROFILE Available from: Gang Yang Retrieved on: 08 October 2015 Algebra Colloquium 20 : 4 (2013) 601{612 Algebra Colloquium °c 2013 AMSS CAS & SUZHOU UNIV 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 Classi¯cation: primary 16D40, 16D50; secondary 16E05, 16E65 Keywords: FP-injective modules, Ding-Chen rings, Ding projective modules, Ding injec- tive 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 di®erent 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 ¯rst 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 ® : M ! C = Coker(f) can be lifted to ¯ : M ! B, i.e., we have the following completed commutative diagram: M ¯ ® f ~ ² 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], StenstrÄomintroduced 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 ¯nitely presented R-modules A. More generally, the FP- injective dimension of an R-module B is de¯ned to be the least integer n ¸ 0 such that Extn+1(A; B) = 0 for all ¯nitely presented R-modules A. The FP-injective dimension of B is denoted by FP-id(B) and equals 1 if no such n above exists. FC rings, as the coherent version of quasi-Frobenius rings where Noetherian is re- placed 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]. De¯nition 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 in¯nite product of ¯elds, 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 ¯nite group (see [1]). Given a class H of R-modules, we will denote the class of R-modules X sat- isfying Ext1(H; X) = 0 (respectively, Ext1(X; H) = 0) for every H 2 H by H? (respectively, ?H). Following [8], we give the following de¯nitions: De¯nition 1.2. A pair of classes of R-modules (A; B) is said to be a cotorsion pair if A? = B and ?B = A. De¯nition 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 2 B and A; Ae 2 A. De¯nition 1.4. Let H be a class of R-modules and X an R-module. A homo- morphism f : X ! H is called an H-preenvelope if H 2 H and the abelian group homomorphism Hom(f; H0) : Hom(H; H0) ! Hom(X; H0) is surjective for each H0 2 H. An H-preenvelope f : X ! H is called special if Ext1(Coker(f);H0) = 0 for all H0 2 H. Dually, H-precovers and special H-precovers can be de¯ned. 2 Ding Modules over General Rings De¯nition 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 De¯nition 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 speci¯ed.
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