Semidualizing Module and Gorenstein Homological Dimensions

Abstract

Let $C$ be a semidualizing module over any commutative ring $R$. We investigate the semidualizing module $C$ with finite injective dimension. In particular, we obtain some equivalent characterizations of $C$ under the trivial extension of $R$ by $C$. Moreover, we get that the supremum of the $C$-Gorenstein projective dimensions of all $R$-modules and the supremum of the $C$-Gorenstein injective dimensions of all $R$-modules are equal. Hence the $C$-Gorenstein global dimension of the ring $R$ is definable. At last, we consider the weak $C$-Gorenstein global dimension.