A note on a graph related to 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.  ‎This article is inspired by the work done on the comaximal ideal graph of a commutative ring‎. ‎Let R be a ring‎.  ‎We associate an undirected graph to R denoted by \mathcal{G}(R)‎,  ‎whose vertex set is the set of all proper ideals I of R such that I\not\subseteq J(R)‎, ‎where J(R) is the Jacobson radical of R  and distinct vertices I۱‎, ‎I۲are adjacent in \mathcal{G}(R) if and only if I۱∩ I۲ = I۱I۲‎.  ‎The aim of this article is to study the interplay between the graph-theoretic properties of \mathcal{G}(R) and the ring-theoretic properties of R.

Keywords:

Comaximal ideal graph of a commutative ring , Complete graph , von Neumann regular ring , bipartite graph , Clique number