A Criterion for Dualizing Modules

Abstract

We establish a characterization of dualizing modules among semidualizing modules. Let $R$ be a finite dimensional commutative Noetherian ring with identity and $C$ a semidualizing $R$-module. We show that $C$ is a dualizing $R$-module if and only if $\mathrm{Tor}_i^R(E,E')$ is $C$-injective for all $C$-injective $R$-modules $E$ and $E'$ and all $i\geq 0$.