Document Type: Original Manuscript

Author

Department of Mathematics, Shahrekord University, P.O. Box 115, Shahrekord, Iran.

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(RM), such that when M=R, Γ(RM) coincide with the zero-divisor graph of R. Many well-known results by D.F. Anderson and P.S. Livingston have been generalized for Γ(RM). We Will show that Γ(RM) is connected with
diam Γ(RM)≤ 3 and if Γ(RM) contains a cycle, then Γ(RM)≤4. We will also show that Γ(RM)=Ø if and only if M is a
prime module. Among other results, it is shown that for a reduced module M satisfying DCC on cyclic submodules,
gr (Γ(RM))=∞ if and only if Γ(RM) is a star graph. Finally, we study the zero-divisor graph of free
R-modules. 

Keywords