Rings with finite Gorenstein injective dimension

Source abstract

In this paper we prove that for any associative ring R R , and for any left R R -module M M with finite projective dimension, the Gorenstein injective dimension G i d R M \mathrm {Gid}_R M equals the usual injective dimension i d R M \mathrm {id}_R M . In particular, if G i d R R \mathrm {Gid}_R R is finite, then also i d R R \mathrm {id}_R R is finite, and thus R R is Gorenstein (provided that R R is commutative and Noetherian).