On the cofinite modules with respect to an ideal of cohomological dimension not exceeding one

Abstract

Let $R$ be a commutative Noetherian ring and $I$ be an ideal of $R$ with $\mathrm{cd}(I,R)\leq 1$. For an $R$-module $M$, we introduce a class of prime ideals, say $\overline{\mathrm{Ass}}_R \ M$, as the set of all prime ideals $\mathfrak{p}$ of $R$ such that $\mathrm{Ann}_R (0 :_M \mathfrak{p})$ = $\mathfrak{p}$. We show that if $R$ is a Noetherian complete local ring and $M$ is an $I$-cofinite $R$-module, then $\overline{\mathrm{Ass}}_R \ M$ is finite. Also, we prove that for each $I$-cofinite $R$-module $M$, $I\not\subseteq \cup_{\mathfrak{p} \in \Lambda_R(I,M)}\ \mathfrak{p}$, where $\Lambda_R(I,M)$ is the set of all maximal elements of $\overline{\mathrm{Ass}}_R \ M\setminus V(I)$ with respect to inclusion. Subsequently, for each $a \in I$, the $R$-module $(0 :_M a)$ is finitely generated if and only if $a \not\in \cup_{\mathfrak{p}\in\Lambda_R(I,M)} \ \mathfrak{p}$.