Some results on a supergraph of the comaximal ideal graph of a commutative ring

The rings considered in this article are commutative with identity which admit at least two maximal ideals‎. ‎We denote the set of all maximal ideals of a ring $R$ by $Max(R)$ and we denote the Jacobson radical of $R$ by $J(R)$‎. ‎Let $R$ be a ring such that $|Max(R)|\geq 2$‎. ‎Let $\mathbb{I}(R)$ denote the set of all proper ideals of $R$‎. ‎In this article‎, ‎we associate an undirected graph denoted by $\mathbb{INC}(R)$ with a subcollection of ideals of $R$ whose vertex set is $\{I\in \mathbb{I}(R)|I\not\subseteq J(R)\}$ and two distinct vertices $I_{1}‎, ‎I_{2}$ are adjacent in $\mathbb{INC}(R)$ if and only if $I_{1}\not\subseteq I_{2}$ and $I_{2}\not\subseteq I_{1}$ (that is‎, ‎$I_{1}$ and $I_{2}$ are not comparable under the inclusion relation)‎. ‎The aim of this article is to investigate the interplay between the graph-theoretic properties of $\mathbb{INC}(R)$ and the ring-theoretic properties of $R$.

Abstract licence: CC BY-SA

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