A Note on Co-Maximal Ideal Graph of Commutative Rings

TL;DR

This paper studies the structure of co-maximal ideal graphs of commutative rings, classifies when these graphs are planar, and answers a specific question about infinite star graphs, advancing understanding of ring-graph relationships.

Contribution

It classifies all rings with planar co-maximal ideal graphs and confirms that rings with infinite star graphs are isomorphic to a product of a field and a local ring.

Findings

Classified all rings with planar co-maximal ideal graphs.

Confirmed rings with infinite star graphs are products of a field and a local ring.

Provided an affirmative answer to a 2012 posed question.

Abstract

Let $R$ be a commutative ring with unity. The co-maximal ideal graph of $R$, denoted by $\Gamma(R)$, is a graph whose vertices are the proper ideals of $R$ which are not contained in the Jacobson radical of $R$, and two vertices $I_1$ and $I_2$ are adjacent if and only if $I_1 + I_2 = R$. We classify all commutative rings whose co-maximal ideal graphs are planar. In 2012 the following question was posed: If $\Gamma(R)$ is an infinite star graph, can $R$ be isomorphic to the direct product of a field and a local ring? In this paper, we give an affirmative answer to this question.