Faltings' annihilator theorem and almost Cohen-Macaulay rings

Faltings' annihilator theorem is an important result in local cohomology theory. Recently, Doustimehr and Naghipour generalized the Falitings' annihilator theorem. They proved that if $R$ is a homomorphic image of a Gorenstein ring, then $f_\mathfrak{a}^\mathfrak{b}(M)_n = \lambda_\mathfrak{a}^\mathfrak{b}(M)_n$, where $f_\mathfrak{a}^\mathfrak{b}(M)_n := \inf\{i \in \mathbb{N} \mid \operatorname{dim}{\operatorname{Supp}(\mathfrak{b}^t H_\mathfrak{a}^i(M))} \geq n \text{ for all } t\in \mathbb{N}\}$ and $\lambda_\mathfrak{a}^\mathfrak{b}(M)_n := \inf\{\lambda_{\mathfrak{a} R_\mathfrak{p}}^{\mathfrak{b} R_\mathfrak{p}}(M_\mathfrak{p}) \mid \mathfrak{p}\in\operatorname{Spec}{R} \text{ with } \operatorname{dim}{R/\mathfrak{p}} \geq n\}$. In this paper, we study the relation between $f_\mathfrak{a}^\mathfrak{b}(M)_n$ and $\lambda_\mathfrak{a}^\mathfrak{b}(M)_n$, and prove that if $R$ is an almost Cohen-Macaulay ring, then $f_\mathfrak{a}^\mathfrak{b}(M)_n \geq \lambda_\mathfrak{a}^\mathfrak{b}(M)_n - \operatorname{cmd}{R}$. Using this result, we prove that if $R$ is a homomorphic image of a Cohen-Macaulay ring, then $f_\mathfrak{a}^\mathfrak{b}(M)_n = \lambda_\mathfrak{a}^\mathfrak{b}(M)_n$.