On the Unit Graph of a Noncommutative Ring

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 $n\in \mathbb{N}$ 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.

6 pages. To appear in Algebra Colloquium