Document Type : Original Manuscript

Authors

1 Department of Mathematics, Roudbar Branch, Islamic Azad University, Roudbar, Iran.

2 Department of Mathematics, Imam Khomeini International University, P.O. Box 34149-1-6818, Qazvin, Iran.

Abstract

‎Let $R$ be a commutative ring and $Z(R)$ be the set of its zero-divisors‎.
‎The annihilator graph of $R$‎, ‎denoted by $AG(R)$ is a simple undirected graph whose vertex‎
‎set is $Z(R)^*$‎, ‎the set of all nonzero zero-divisors of $R$‎, ‎and two distinct vertices $x$ and‎
‎$y$ are adjacent if and only if ${\rm ann}_R(xy)\neq {\rm ann}_R(x)\cup {\rm ann}_R(y)$‎.
‎In this paper‎, ‎perfectness of the annihilator graph for some classes of rings is investigated‎.
‎More precisely‎, ‎we show that if $R$ is an Artinian ring‎, ‎then $AG(R)$ is perfect‎.

Keywords

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