THE ZERO-DIVISOR GRAPH OF A MODULE
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Abstract:
Let R be a commutative ring with identity and M an R-module. In this paper, we associate a graph to M, sayΓ(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 withdiam 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 aprime 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 freeR-modules.
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Journal title
volume 4 issue 2
pages 155- 171
publication date 2017-01-01
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