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 Z R (M) Ann R (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 Z R (M) r(Ann R (M)) if and only if R = ⊕Z 2 ⊕R ′ and M = Z 2 ⊕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) Ass R (M) = {m 1 , m 2 }, where m 1 , m 2 are maximal ideals of R. (ii) Ass R (M) = {p}, where p 2 ⊆ Ann R (M). (iii) Ass R (M) = {p}, where p 3 ⊆ Ann R (M).

A Generalization of the Essential Graph for Modules Over Commutative Rings

“…In this section we will study the relations between the zero-divisor graph defined in [11] and the essential graph for modules. xyM = 0.…”

“…The essential graph of R is a variation of the zero-divisor graph that changes the edge condition, and is introduced and studied in [10]. The essential graph of R is a simple undirected graph, denoted by EG(R), with vertex set Z * (R) and two distinct vertices x and y are adjacent if and only if Ann R (xy) is an essential ideal of R. Recently, a lot of research (e.g., [5,7,8,11,12]) has been devoted to the zerodivisor graph of a module (Definition 4.1). Let M be an R-module and let Z(M ) be its set of zero-divisors.…”