A generalization of the zero-divisor graph for modules

Abstract

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)\AnnR(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)\r(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).