A note on co-maximal graph of non-commutative rings

TL;DR

This paper corrects a previous proof about the structure of the co-maximal graph of commutative rings and extends the results to non-commutative rings, providing a more accurate understanding of their algebraic graph properties.

Contribution

It provides a correct proof of a key result relating the n-partiteness of a graph to the number of maximal ideals, and generalizes these findings to non-commutative rings.

Findings

Corrected the proof of the n-partite property of the graph

Extended results to non-commutative rings

Enhanced understanding of co-maximal graph structure

Abstract

Let $R$ be a ring with unity. The graph $\Gamma(R)$ is a graph with vertices as elements of $R$, where two distinct vertices $a$ and $b$ are adjacent if and only if $Ra+Rb=R$. Let $\Gamma_2(R)$ is the subgraph of $\Gamma(R)$ induced by the non-unit elements. H.R. Maimani et al. [H.R. Maimani et al., Comaximal graph of commutative rings, J. Algebra $319$ $(2008)$ $1801$-$1808$] proved that: ``If $R$ is a commutative ring with unity and the graph $\Gamma_2(R)\backslash J(R)$ is $n$-partite, then the number of maximal ideals of $R$ is at most $n$." The proof of this result is not correct. In this paper we present a correct proof for this result. Also we generalize some results given in the aforementioned paper for the non-commutative rings.