Journal of the
Korean Mathematical Society JKMS

Let $\mathcal{E}$ be an injectively resolving class of left $R$-modules and $n\geq 1$ be an integer. In this paper, we introduce and study $n$-strongly $\mathcal{E}$-Gorenstein injective and flat modules, which are common generalizations of strongly Gorenstein injective and flat modules \cite{BM07, YL08}, $n$-strongly Gorenstein injective and flat modules \cite{BM09, ZH11}, respectively. Then we discuss some properties of these modules and show how the property of being an $n$-strongly $\mathcal{E}$-Gorenstein injective and flat module can be inherited by its direct summands under certain condition. Finally, we consider the connections between $n$-strongly $\mathcal{E}$-Gorenstein injective and flat modules. As applications, some known results are obtained as corollaries.

Keywords: Injectively resolving class, $n$-strongly $\mathcal{E}$-Gorenstein injective module, $n$-strongly $\mathcal{E}$-Gorenstein flat module

MSC numbers: 18G25, 16E05, 16E30

Supported by: This work was partially supported by National Natural Science Foundation of China (No.~12371038 and 12061001), the Sichuan Science and Technology Program (No. 2024NSFSC0417 and 2023ZYD0005) and the Scientific Research Foundation of CUIT (No. KYTD202331).