On depth of modules in an ideal

Let $R$ be a commutative Noetherian ring, $I$ an ideal of $R$ and $M$ a finitely generated $R$-module with ${\dim }_R(M)=d$. Denote by ${\text{depth}}_R(I,M)$ the depth of $M$ in $I$. In \cite{HT}, C. Huneke and V. Trivedi proved that if $R$ is a quotient of a regular ring then there exists a finite subset ${\mathrm{\Lambda }}_M$ of $\text{Spec}(R)$ such that $${\text{depth}}_R(I,M)=\{{\text{depth}}_{R_{\text{p}}}(M_{\text{p}})+\text{docao}((I+\text{p})\mathrm{/}\text{p}\left)\right\}\mathrm{.}$$ Denote by ${\text{Psupp}}_R^i(M)=\{\mathfrak{p}\in \text{Spec}(R)\mid H_{\mathfrak{p}{R}_{\mathfrak{p}}}^{i-\dim (R\mathrm{/}\mathfrak{p})}(M_{\mathfrak{p}})\mathrm{\ne }0\}$ the $i$-th pseudo support of $M$ defined by M. Brodmann and R. Y. Sharp \cite{BS1}. In this paper, we prove that if ${\text{Psupp}}_R^i(M)$ is closed for all $i\le d$ then the above formula of ${\text{depth}}_R(I,M)$ holds true, where ${\mathrm{\Lambda }}_M=\min {\text{Psupp}}_R^i(M)$. In particular, if $R$ is a quotient of a Cohen-Macaulay local ring then ${\mathrm{\Lambda }}_M=\min \text{Var}({\text{Ann}}_R(H_{\text{m}}^i(M)))$. We also give some examples to clarify the results.