Document Type : Original Manuscript

Authors

Department of Mathematical Sciences, Shahrekord University, P.O.Box 115, Shahrekord, Iran.

Abstract

Let $R$ be a ring (not necessarily commutative) with nonzero identity. We define $\Gamma(R)$ to be the graph with vertex set $R$ in which two distinct vertices $x$ and $y$ are adjacent if and only if there exist unit elements $u,v$ of $R$ such that $x+uyv$ is a unit of $R$. In this paper, basic properties of $\Gamma(R)$ are studied. We investigate connectivity and the girth of $\Gamma(R)$, where $R$ is a left Artinian ring. We also determine when the graph $\Gamma(R)$ is a cycle graph. We prove that if $\Gamma(R)\cong\Gamma(M_{n}(F))$ then $R\cong M_{n}(F)$, where $R$ is a ring and $F$ is a finite field. We show that if $R$ is a finite commutative semisimple ring and $S$ is a commutative ring such that $\Gamma(R)\cong\Gamma(S)$, then $R\cong S$. Finally, we obtain the spectrum of $\Gamma(R)$, where $R$ is a finite commutative ring.

Keywords