A generalization of the zero-divisor graph for modules

Abstract

Let R be a commutative ring with identity and M an R- module. In this paper, we associate a graph to M, say Γ(M), such that when M = R, Γ(M) is exactly the classic zero-divisor graph. Many well- known results by D. F. Anderson and P. S. Livingston, in [5], and by D. F. Anderson and S. B. Mulay, in [6], have been generalized for Γ(M) in the present article. We show that Γ(M) is connected with diam(Γ(M)) ≤ 3. We also show that for a reduced module M with Z(M)* ≠ M \ {0}, gr(Γ(M)) = ∞ if and only if Γ(M) is a star graph. Furthermore, we show that for a finitely generated semisimple R-module M such that its homogeneous components are simple, x, y ∈ M \ {0} are adjacent if and only if xR{n-ary intersection}yR = (0). Among other things, it is also observed that Γ(M) = ∅ if and only if M is uniform, ann(M) is a radical ideal, and Z(M)* ≠ M \ {0}, if and only if ann(M) is prime and Z(M)* ≠ M \ {0}. © 2014 The Korean Mathematical Society.