Projectively coresolved Gorenstein flat and ding projective modules

Abstract

We give necessary and sufficient conditions in order for the class of projectively coresolved Gorenstein flat modules, PGF , (respectively that of projectively coresolved Gorenstein B flat modules, PGF B ) to coincide with the class of Ding projective modules (DP). We show that PGF = DP if and only if every Ding projective module is Gorenstein flat. This is the case if the ring R is coherent for example. We include an example to show that the coherence is a sufficient, but not a necessary condition in order to have PGF = DP. We also show that PGF = DP over any ring R of finite weak Gorenstein global dimension (this condition is also sufficient, but not necessary). We prove that if the class of Ding projective modules, DP, is covering then the ring R is perfect. And we show that, over a coherent ring R, the converse also holds. We also give necessary and sufficient conditions in order to have PGF = GP, where GP is the class of Gorenstein projective modules.2010 MSC: 16E05, 16E10.

$(\mathcal{F},\mathcal{A})$-Gorenstein flat homological dimensions

“…Finally in Section 6 we give examples of the relative Hom-balance and several examples of (L, A)-Gorenstein rings where the − ⊗ R − occurs and we see how one of this examples recover the well-know result on a Ding-Chen ring [27,Theorem 3.23]. Finally in Corollary 6.4 and Proposition 6.5 we answer a question proposed by A. Iacob [22], namely; When it is true that the Ding projective modules and the Gorenstein projective modules coincide?. We also give conditions in Proposition 6.7 for other Gorenstein classes to coincide with the Gorenstein projective R-modules.…”

“…From the hypothesis of the previous results we see that it is important to know when the class GF (F ,A) is closed under extensions, this has been studied recently. From [22,Proposition 7] we know that this occurs when the class A is semi-definable and…”

“…By definition and without conditions on the ring R it is known that DP(R) ⊆ GP(R). Recently A. Iacob [22] has considered this question as part of the tools developed around the study of class GF (P,B) (R) where some occasions B corresponds to a definable class. It is known that over a Ding-Chen ring R the class of Ding-projective R-modules coincides with the class of Gorenstein projective R-modules [17, Theorem 1.1], here we will see that the above also occurs when the finitistic projective dimension of R in finite.…”

How to construct Gorenstein projective modules relative to complete duality pairs over Morita rings

“…with M ∼ = Ker(P 0 → P 0 ) such that Hom R (P, F ) is exact for any flat left R-module F . For more details on Ding projective modules, we refer to [4,11,21,25,16]. The following is the main result of this section which contains Theorem 1.2 in the introduction.…”

FP-injective dimensions and Gorenstein homology

“…This means every flat -module by Corollary 3.4. Therefore, it follows by [20, Lemma 4] that any Gorenstein projective -module is Ding projective. The last assertion follows then by Corollary 3.5.…”