On the Unit Graph of a Noncommutative Ring

TL;DR

This paper generalizes the characterization of the unit graph of a ring, showing it is complete r-partite under certain conditions, and explores properties of rings with finite clique number in their unit graphs.

Contribution

It extends previous results to noncommutative rings, establishing conditions for the unit graph to be complete r-partite and linking finite clique number to ring finiteness.

Findings

$G(R)$ is complete r-partite iff $(R, m)$ is local and $r=|R/m|=2^n$ or $R$ is a finite field.

If $R$ is left Artinian, $2$ is a unit, and $G(R)$ has finite clique number, then $R$ is finite.

Generalizes known results from commutative to noncommutative rings.

Abstract

Let $R$ be a ring (not necessary commutative) with non-zero identity. The unit graph of $R$, denoted by $G(R)$, is a graph with elements of $R$ as its vertices and two distinct vertices $a$ and $b$ are adjacent if and only if $a+b$ is a unit element of $R$. It was proved that if $R$ is a commutative ring and $\fm$ is a maximal ideal of $R$ such that $|R/\fm|=2$, then $G(R)$ is a complete bipartite graph if and only if $(R, \fm)$ is a local ring. In this paper we generalize this result by showing that if $R$ is a ring (not necessary commutative), then $G(R)$ is a complete $r$-partite graph if and only if $(R, \fm)$ is a local ring and $r=|R/m|=2^n$, for some $n \in \N$ or $R$ is a finite field. Among other results we show that if $R$ is a left Artinian ring, $2 \in U(R)$ and the clique number of $G(R)$ is finite, then $R$ is a finite ring.