Some Remarks on the Annihilator Graph of a Commutative Ring

Abstract

Let $R$ be a commutative ring with nonzero identity. The annihilator graph of $R$, denoted by $AG(R)$, is the (undirected) graph whose vertex set is the set of all nonzero zero-divisors of $R$ and two distinct vertices $x$ and $y$ are adjacent if and only if $\ann_R(xy)\neq {\rm ann}_R(x)\cup {\rm ann}_R(y)$. We investigate the interplay between ring-theoretic properties of $R$ and graph-theoretic properties of $AG(R)$. We study the relation between two graphs $\Gamma(R)$ and $AG(R)$, where $R$ is a non-reduced commutative ring. Also, we completely characterize the rings whose annihilator graphs are complete.