On the intersection graph of ideals of a commutative ring

Abstract

Let $R$ be a commutative ring and $M$ be an $R$-module, and let $I(R{)}^{\ast }$ 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 $I(R{)}^{\ast }$, and two distinct vertices $I$ and $J$ are adjacent if and only if $IM\cap JM\ne 0$. 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 $n,m\ge 2$ 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.

Cited by in corpus (1)

• Planar, outerplanar and ring graph of the intersection graph