STRONGLY E-GORENSTEIN INJECTIVE AND FLAT MODULES

Abstract

Let $\mathcal{ℰ}$ be an injectively resolving subcategory of left $R$-modules. We study a particular case of $\mathcal{ℰ}$-Gorenstein injective and flat modules, called strongly $\mathcal{ℰ}$-Gorenstein injective and flat modules, respectively. We prove that a module is $\mathcal{ℰ}$-Gorenstein injective if and only if it is a direct summand of a strongly $\mathcal{ℰ}$-Gorenstein injective module, and every $\mathcal{ℰ}$-Gorenstein flat module is a direct summand of a strongly $\mathcal{ℰ}$-Gorenstein flat module. Then we show the property of being a strongly $\mathcal{ℰ}$-Gorenstein injective (resp. flat) module can be inherited by its direct summands under certain condition. The connections between (strongly) $\mathcal{ℰ}$-Gorenstein injective and flat modules are also discussed. Finally, we investigate FC rings in terms of strongly $\mathcal{ℰ}$-Gorenstein injective and flat modules.