Gorenstein injective modules and Ext.

Abstract

The aim of this paper is to characterize $n$-Gorenstein rings in terms of Gorenstein injective modules and the Ext functor. We will show that if $R$ is a left and right noetherian ring and $n$ is a positive integer, then $R$ is $n$-Gorenstein if and only if $M$ being Gorenstein injective means that Ext$^{1}$ $(L, M )=0$ for all countably generated $R$-modules $L$ of projective dimension at most $n$. In particular, if $R$ is $n$-Gorenstein, then an $R$-module $M$ is Gorenstein injective if and only if it is $U$-Gorenstein injective whenever $U$ is a free $R$-module with a countable base.