Projectively coresolved Gorenstein flat and Ding projective modules

We give necessary and sufficient conditions in order for the class of projectively coresolved Gorenstein flat modules, $\mathcal{P}\mathcal{G}\mathcal{F}$, (respectively that of projectively coresolved Gorenstein $\mathcal{B}$ flat modules, ${\mathcal{P}\mathcal{G}\mathcal{F}}_{\mathcal{B}}$) to coincide with the class of Ding projective modules ($\mathcal{D}\mathcal{P})$. We show that $\mathcal{P}\mathcal{G}\mathcal{F}=\mathcal{D}\mathcal{P}$ 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 $\mathcal{P}\mathcal{G}\mathcal{F}=\mathcal{D}\mathcal{P}$. We also show that $$\mathcal{P}\mathcal{G}\mathcal{F}$ = $\mathcal{D}\mathcal{P}$$overanyring$R$ of finite weak Gorenstein global dimension (this condition is also sufficient, but not necessary). We prove that if the class of Ding projective modules, $\mathcal{D}\mathcal{P}$, 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 $\mathcal{P}\mathcal{G}\mathcal{F}=\mathcal{G}\mathcal{P}$, where $\mathcal{G}\mathcal{P}$ is the class of Gorenstein projective modules.