Auslander-Reiten conjecture and finite injective dimension of Hom

For a finitely generated module $ M $ over a commutative Noetherian ring $R$, we settle the Auslander-Reiten conjecture when at least one of ${rm Hom}_R(M,R)$ and ${rm Hom}_R(M,M)$ 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.