Document Type : Original Manuscript


Department of Mathematics, Imam Khomeini International University, P.O.Box 34149-16818, Qazvin, Iran.



‎Let $R$ be a commutative ring and $M$ be an $R$-module‎. ‎The‎
‎annihilator graph of $M$‎, ‎denoted by $AG(M)$ is a simple undirected‎
‎graph associated to $M$ whose the set of vertices is‎
‎$Z_R(M) \setminus {\rm Ann}_R(M)$ and two distinct vertices $x$ and‎
‎$y$ are adjacent if and only if ${\rm Ann}_M(xy)\neq {\rm‎
‎Ann}_M(x) \cup {\rm Ann}_M(y)$‎. ‎In this paper‎, ‎we study the‎
‎diameter and the girth of $AG(M)$ and we characterize all modules‎
‎whose annihilator graph is complete‎. ‎Furthermore‎, ‎we look for the‎
‎relationship between the annihilator graph of $M$ and its zero-divisor‎


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