The annihilator graph of modules over commutative rings

Abstract

Let $M$ be a module over a commutative ring $R$, $Z_{*}(M)$ be its set of weak zero-divisor elements, andif $min M$, then let $I_m=(Rm:_R M)={rin R : rMsubseteq Rm}$. The annihilator graph of $M$ is the (undirected) graph$AG(M)$ with vertices $tilde{Z_{*}}(M)=Z_{*}(M)setminus {0}$, and two distinct vertices $m$ and $n$ are adjacent if andonly if $(0:_R I_{m}I_{n}M)neq (0:_R m)cup (0:_R n)$. We show that $AG(M)$ is connected with diameter at most two and girth at mostfour. Also, we study some properties of the zero-divisor graph of reduced multiplication-like $R$-modules.