A generalization of Gorenstein injective modules

Abstract

Let $R$ be an associative ring with identity and $M$ be a left $R$-module. In this paper, we introduce $M$-Gorenstein injective modules as a generalization of Gorenstein injective modules. We verify some properties of $M$-Gorenstein injective modules analogous to those holding for Gorenstein injective modules. There is an interesting theorem in classical homological algebra which asserts that $R$ is a Noetherian ring if and only if the class of injective modules over $R$ is closed under arbitrary direct sum. Our goal in this paper is to investigate the $M$-Gorenstein injective counterpart of this fact. If the class of $M$-Gorenstein injective modules over $R$ is closed under arbitrary direct sum, then $R$ will be a Noetherian ring. Also, it has been proved that in the special case $M=R$, when $R$ is a commutative Noetherian ring with a dualizing complex, then the class of $R$-Gorenstein injective modules is closed under arbitrary direct sum. In the main theorem of this paper, we prove the general case of this result. More precisely, we show that for any left $R$-module $M$ over a Noetherian ring $R$ in which every $R$-module has finite $M$-Gorenstein injective dimension, the class of $M$-Gorenstein injective modules is closed under arbitrary direct sum.