The annihilator graph of modules over commutative rings

Let M be a module over a commutative ring R, Z_{*}(M) be its set of weak zero-divisor elements, andif m\in M, then let I_m=(Rm:_R M)=\{r\in R : rM\subseteq Rm\}. The annihilator graph of M is the (undirected) graphAG(M) with vertices \tilde{Z_{*}}(M)=Z_{*}(M)\setminus \{۰\}, and two distinct vertices m and n are adjacent if andonly if (۰:_R I_{m}I_{n}M)\neq (۰:_R m)\cup (۰:_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.

Keywords:

Annihilator graph , reduced module , multiplication-like module