THE ANNIHILATOR GRAPH FOR MODULED OVER COMMUTATIVE RINGS

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.