On the intersection graph of ideals of a commutative ring

TL;DR

This paper introduces and analyzes the $M$-intersection graph of ideals of a commutative ring, exploring its properties, such as diameter, girth, and conditions for perfection, with specific focus on modules and rings like $Z_m$.

Contribution

It defines the $M$-intersection graph of ideals, determines its key properties for multiplication modules, and characterizes when certain graphs are perfect or weakly perfect.

Findings

Diameter and girth are determined for multiplication modules.

If the clique number is finite and the module is faithful, then the ring is semilocal.

Conditions for the $G_n(Z_m)$ graph to be perfect or weakly perfect are established.

Abstract

Let $R$ be a commutative ring and $M$ be an $R$-module, and let $I(R)^*$ be the set of all non-trivial ideals of $R$. The $M$-intersection graph of ideals of $R$, denoted by $G_M(R)$, is a graph with the vertex set $I(R)^*$, and two distinct vertices $I$ and $J$ are adjacent if and only if $IM\cap JM\neq 0$. For every multiplication $R$-module $M$, the diameter and the girth of $G_M(R)$ are determined. Among other results, we prove that if $M$ is a faithful $R$-module and the clique number of $G_M(R)$ is finite, then $R$ is a semilocal ring. We denote the $\mathbb{Z}_n$-intersection graph of ideals of the ring $\mathbb{Z}_m$ by $G_n(\mathbb{Z}_m)$, where $n,m\geq 2$ are integers and $\mathbb{Z}_n$ is a $\mathbb{Z}_m$-module. We determine the values of $n$ and $m$ for which $G_n(\mathbb{Z}_m)$ is perfect. Furthermore, we derive a sufficient condition for $G_n(\mathbb{Z}_m)$ to be weakly perfect.