Gorenstein injective modules and Ext. (Q2565489)

An associative ring with identity \(R\) is called \(n\)-Gorenstein when \(R\) is left and right Noetherian and has injective dimension at most \(n\) on either side. The purpose of this paper is to characterize these rings in terms of Gorenstein injective modules and the Ext functor. It is shown that if \(R\) is left and right Noetherian, then \(R\) is \(n\)-Gorenstein if and only if \(\text{Ext}_R^1(L,M)=0\) for all \(R\)-modules \(L\) of projective dimension at most \(n\) implies that \(M\) is Gorenstein injective. Furthermore, this characterization still holds if the modules \(L\) are restricted to be countably generated with projective dimension at most \(n\). The last part of the paper explores similar conditions for \(U\)-Gorenstein injective modules, where \(U\) is an \(R\)-module.