On relative Gorenstein homological dimensions with respect to a dualizing module

Abstract

Let $R$ be a commutative Noetherian ring. The aimof this paper is studying the properties of relative Gorensteinmodules with respect to a dualizing module. It is shown that everyquotient of an injective module is $G_C$-injective, where $C$ isa dualizing $R$-module with $i{d}_R(C)\le 1$. We also provethat if $C$ is a dualizing module for a local integral domain,then every $G_C$-injective $R$-module is divisible. In addition,we give a characterization of dualizing modules via relativeGorenstein homological dimensions with respect to a semidualizingmodule.