Razi University

Let  R  e a Noetherian local ring, and let  C  e a semidualizing  R-module. In this paper, we present someresults concerning the vanishing of  Ext  and finite injective dimension of  Hom. Additionally, we extendthese results in terms of finite  C-injective dimension of  Hom. We also investigate the consequencesof some of these extensions in the case where  R  is Cohen–Macaulay and  C  is a canonical module forR. Furthermore, we provide positive answers to the Auslander–Reiten conjecture for finitely generatedR-module  M  uch that  IC  -idR(HomR(M, R)) < ?or  M ?AC  (R)  with  IC  -idR(HomR(M, M)) < ?.Moreover, we derive a number of criteria for a semidualizing  R-module  C  to be a canonical module for  Rin terms of the vanishing of Ext and the finite C-injective dimension of Hom.