STRONGLY E-GORENSTEIN INJECTIVE AND FLAT MODULES

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