On $\mathrm{Ext}$-finite modules, quasi-injective dimension and width of modules

Let $(R,\mathfrak{m},k)$ be a commutative Noetherian local ring. It is well-known that if $M$ is a finitely generated $R$-module of finite quasi-injective dimension, then ${\mathrm{qid}}_{R}M=\mathrm{depth}R$. In this paper, we demonstrate that under the weaker condition that $M$ is $\mathrm{Ext}$-finite and has finite quasi-injective dimension, the equality ${\mathrm{qid}}_{R}M=0$ holds if and only if ${\mathrm{Ext}}_R^{i>0}(R\mathrm{/}(\mathit{x}),M)=0$, where $\mathit{x}\in \mathfrak{m}$ is a maximal $R$-sequence and if ${\mathrm{qid}}_{R}M\mathrm{\ne }0$, we show then that ${\mathrm{qid}}_{R}M=\sup \{i:{\mathrm{Ext}}_R^i(R\mathrm{/}(\mathit{x}),M)\mathrm{\ne }0\}$. Also, we show that if $R$ is a Cohen-Macaulay local ring and $M$ is an $\mathrm{Ext}$-finite $R$-module of finite quasi-injective dimension, then $\mathrm{depth}R={\mathrm{qid}}_{R}M+\inf \{i:{\mathrm{Tor}}_i^R(k,M)\mathrm{\ne }0\}$, provided that $\inf \{i:{\mathrm{Tor}}_i^R(k,M)\mathrm{\ne }0\}<\mathrm{\infty }$.