On the extended zero divisor graph of commutative rings

Abstract

In this paper we present a new graph that is closely related to the classical zero-divisor graph. In our case two nonzero distinct zero divisors $x$ and $y$ of a commutative ring $R$ are adjacent whenever there exist two nonnegative integers $n$ and $m$ such that $x^ny^m=0$ with $x^n\neq 0$ and $y^m\neq 0$. This yields an extension of the classical zero divisor graph $\Gamma(R)$ of $R$, which will be denoted by $\overline{\Gamma}(R)$. First we distinguish when $\overline{\Gamma}(R)$ and $\Gamma(R)$ coincide. Various examples in this context are given. We show that if $\overline{\Gamma}(R) \not=\Gamma(R)$, then $\overline{\Gamma}(R)$ must contain a cycle. We also show that if $\overline{\Gamma}(R) \not=\Gamma(R)$ and $\overline{\Gamma}(R)$ is complemented, then the total quotient ring of $R$ is zero-dimensional. Among other things, the diameter and girth of $\overline{\Gamma}(R)$ are also studied.