THE ZERO-DIVISOR GRAPH OF A MODULE

Abstract

Let R be a commutative ring with identity and M an R-module. In this paper, we associate a graph to M, say<br />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<br />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<br />prime module. Among other results, it is shown that for a reduced module M satisfying DCC on cyclic submodules,<br />gr (Γ(RM))=∞ if and only if Γ(RM) is a star graph. Finally, we study the zero-divisor graph of free<br />R-modules.