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, $\mathcal{PGF}$, (respectively that of projectively coresolved Gorenstein $\mathcal{B}$ flat modules, $\mathcal{PGF}_{\mathcal{B}}$) to coincide with the class of Ding projective modules ($\mathcal{DP})$. We show that $\mathcal{PGF} = \mathcal{DP}$ 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 $\mathcal{PGF} = \mathcal{DP}$. We also show that $\mathcal{PGF} = \mathcal{DP}$ 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 $\mathcal{PGF} = \mathcal{GP}$, where $\mathcal{GP}$ is the class of Gorenstein projective modules.
- Publication:
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arXiv e-prints
- Pub Date:
- January 2020
- DOI:
- 10.48550/arXiv.2001.09234
- arXiv:
- arXiv:2001.09234
- Bibcode:
- 2020arXiv200109234I
- Keywords:
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- Mathematics - Rings and Algebras;
- 16E05;
- 16E10
- E-Print:
- 17 pages, Communications in Algebra, accepted