On the diameter of the graph ΓAnn(M)(R)

For a commutative ring R with identity, the ideal-based zero- divisor graph, denoted by ΓI(R), is the graph whose vertices are {x ∊ R \ I | xy∊ I for some y ∊ R \ I}, and two distinct vertices x and y are adjacent if and only if xy ∊ I. In this paper, we investigate an annihilator ideal-based zero-divisor graph, denoted by ΓAnn(M)(R), by replacing the ideal I with the annihilator ideal Ann(M) for an R-module M. We also study the relationship between the diameter of ΓAnn(M)(R) and the minimal prime ideals of Ann(M). In addition, we determine when ΓAnn(M)(R) is complete. In particular, we prove that for a reduced R-module M, ΓAnn(M)(R) is a complete graph if and only if R≅ Z2 × Z2 and M ≅ M1 × M2 for M1 and M2 nonzero Z2-modules.