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, $\mathcal{P}\mathcal{GF}$, (respectively that of projectively coresolved Gorenstein $\mathcal{B}$ flat modules, ${\mathcal{P}\mathcal{GF}}_{\mathcal{B}}$) to coincide with the class of Ding projective modules ($\mathcal{DP})$. We show that $\mathcal{P}\mathcal{GF}=\mathcal{DP}$ 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 $\mathcal{P}\mathcal{GF}=\mathcal{DP}$. We also show that $\mathcal{P}\mathcal{GF}=\mathcal{DP}$ 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, $\mathcal{DP}$, 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 $\mathcal{P}\mathcal{GF}=\mathcal{G}\mathcal{P}$, where $\mathcal{G}\mathcal{P}$ is the class of Gorenstein projective modules.

17 pages, Communications in Algebra, accepted