On graphs related to co-maximal ideals of a commutative ring

TL;DR

This paper explores the structure and properties of graphs derived from co-maximal ideals in commutative rings, revealing their core composition and relationships with ring invariants like maximal ideals.

Contribution

It introduces and analyzes the co-maximal graph, its induced subgraph, and a related retract, providing new insights into their structure and connection to ring properties.

Findings

Core of the graph is a union of triangles and rectangles.

Vertices are either end vertices or part of the core.

Chromatic and clique numbers equal the number of maximal ideals for non-local rings.

Abstract

This paper studies the co-maximal graph $\Om(R)$, the induced subgraph $\G(R)$ of $\Om(R)$ whose vertex set is $R\setminus (U(R)\cup J(R))$ and a retract $\G_r(R)$ of $\G(R)$, where $R$ is a commutative ring. We show that the core of $\G(R)$ is a union of triangles and rectangles, while a vertex in $\G(R)$ is either an end vertex or a vertex in the core. For a non-local ring $R$, we prove that both the chromatic number and clique number of $\G(R)$ are identical with the number of maximal ideals of $R$. A graph $\G_r(R)$ is also introduced on the vertex set $\{Rx|\,x\in R\setminus (U(R)\cup J(R))\}$, and graph properties of $\G_r(R)$ are studied.