A generalization of the zero-divisor graph for modules

Let R be a commutative ring and M a Noetherian R-module. The zero-divisor graph of M , denoted by Γ(M), is an undirected simple graph whose vertices are the elements of ZR(M)rAnnR(M) and two distinct vertices a and b are adjacent if and only if abM = 0. In this paper, we study diameter and girth of Γ(M). We show that the zero-divisor graph of M has a universal vertex in ZR(M)rr(AnnR(M)) if and only if R = ⊕Z2 ⊕R ′ and M = Z2 ⊕M , where M ′ is an R-module. Moreover, we show that if Γ(M) is a complete graph, then one of the following statements is true: (i) AssR(M) = {m1,m2}, where m1,m2 are maximal ideals of R. (ii) AssR(M) = {p}, where p 2 ⊆ AnnR(M). (iii) AssR(M) = {p}, where p 3 ⊆ AnnR(M).