University of Guilan Journal of Algebra and Related Topics 2345-3931 9 1 2021 06 01 The annihilator graph of modules over commutative rings 93 108 4811 10.22124/jart.2021.18226.1241 EN F. Esmaeili Khalil Saraei Fouman Faculty of Engineering, College of Engineering, University of Tehran, Fouman, Iran. Journal Article 2020 11 21 Let $M$ be a module over a commutative ring $R$, $Z_{*}(M)$ be its set of weak zero-divisor elements, and<br />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<br />$AG(M)$ with vertices $\tilde{Z_{*}}(M)=Z_{*}(M)\setminus \{0\}$, and two distinct vertices $m$ and $n$ are adjacent if and<br />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<br />four. Also, we study some properties of the zero-divisor graph of reduced multiplication-like $R$-modules. https://jart.guilan.ac.ir/article_4811_7cda1806b8113ad6b7aa04c7375a215f.pdf