The annihilator graph of modules over commutative rings

Document Type : Research Paper


Fouman Faculty of Engineering, College of Engineering, University of Tehran, Fouman, Iran.


Let $M$ be a module over a commutative ring $R$, $Z_{*}(M)$ be its set of weak zero-divisor elements, and
if $m\in M$, then let $I_m=(Rm:_R M)=\{r\in R : rM\subseteq Rm\}$. 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 and
only 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 most
four. Also, we study some properties of the zero-divisor graph of reduced multiplication-like $R$-modules.