The Auslander-Reiten Conjecture, Finite $C$-Injective Dimension of $\mathrm{Hom}$, and vanishing of $\mathrm{Ext}$

Let $R$ be a Noetherian local ring, and let $C$ be a semidualizing $R$-module. In this paper, we present some results concerning the vanishing of $\mathrm{Ext}$ and finite injective dimension of $\mathrm{Hom}$. Additionally, we extend these results in terms of finite $C$-injective dimension of $\mathrm{Hom}$. We also investigate the consequences of some of these extensions in the case where $R$ is Cohen-Macaulay and $C$ is a canonical module for $R\mathrm{.}$ Furthermore, we provide positive answers to the Auslander-Reiten conjecture for finitely generated $R$-modules $M$ such that ${\mathcal{I}}_C{\mathrm{-id}}_R({\mathrm{Hom}}_R(M,R))<\mathrm{\infty }$ or $M\in {\mathcal{A}}_C(R)$ with ${\mathcal{I}}_C{\mathrm{-id}}_R({\mathrm{Hom}}_R(M,M))<\mathrm{\infty }$. Moreover, we derive a number of criteria for a semidualizing $R$-module $C$ to be a canonical module for $R$ in terms of the vanishing of $\mathrm{Ext}$ and the finite $C$-injective dimension of $\mathrm{Hom}$.