The annihilator graph of a module

Let R be a commutative ring and M be an R-module, if x 2 M, Ix := fr 2 RjrM Rxg and ann(IxM = fr 2 RjrIxM = 0g. The annihilator graph of module M over ring R is the graph AG(RM) with vertices UMR := fx 2 Mj0 6= Ix 6= Rg and two vertices x and y are adjacent if and only if ann(IxIyM) 6= ann(IxM)[ann(IyM). It follows that if M be a faitful R-module, then each edge of G(RM) is an edge of AG(RM). We show that ifM be amultiplicationmodule then AG(RM) is connected with diameter at most three and with girth at most four provided that AG(RM) has a cycle.