Auslander–Reiten conjecture and finite injective dimension of Hom

Abstract

For a finitely generated module M over a commutative Noetherian ring R, we settle the Auslander–Reiten conjecture when at least one of ${Hom}_R(M,R)$ and ${Hom}_R(M,M)$ has finite injective dimension. A number of new characterizations of Gorenstein local rings are also obtained in terms of vanishing of certain Ext and finite injective dimension of Hom.