THE ZERO-DIVISOR GRAPH OF A MODULE
Publish place: Journal of Algebraic Systems، Vol: 4، Issue: 2
Publish Year: 1396
نوع سند: مقاله ژورنالی
زبان: English
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شناسه ملی سند علمی:
JR_JAS-4-2_006
تاریخ نمایه سازی: 18 تیر 1398
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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Authors
A. Naghipour
Department of Mathematics, Shahrekord University, P.O. Box ۱۱۵, Shahrekord, Iran.