  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.
