Document Type : Original Manuscript
Authors
Department of Mathematics, Imam Khomeini International University, P.O.Box 34149-16818, Qazvin, Iran.
Abstract
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
graph.
Keywords
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