On extension closure of $\mathcal{E}$-Gorenstein flat modules

Abstract

Let $R$ be an arbitrary ring and $\mathcal{E}$ an injectively resolving class of left $R$-modules. We prove that the class of $\mathcal{E}$-Gorenstein flat right $R$-modules is closed under extensions, and hence projectively resolving. This answers an open question in Gao and Zhong [Rocky Mountain J. Math. 54 (2024), 143–160] affirmatively. As a consequence, we get that this class is covering. In addition, we introduce the notion of $\mathcal{E}$-projectively coresolved Gorenstein flat modules, and prove that the class of $\mathcal{E}$-projectively coresolved Gorenstein flat right $R$-modules is projectively resolving and closed under transfinite extensions.