THE ZERO-DIVISOR GRAPH OF A MODULE | ||
| Journal of Algebraic Systems | ||
| مقاله 6، دوره 4، شماره 2، فروردین 2017، صفحه 155-171 اصل مقاله (136.18 K) | ||
| نوع مقاله: Original Manuscript | ||
| شناسه دیجیتال (DOI): 10.22044/jas.2017.858 | ||
| نویسنده | ||
| A. Naghipour* | ||
| Department of Mathematics, Shahrekord University, P.O. Box 115, Shahrekord, Iran. | ||
| چکیده | ||
| 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. | ||
| کلیدواژهها | ||
| Annilhilator؛ diameter؛ girth؛ reduced module؛ zero-divisor graph | ||
|
آمار تعداد مشاهده مقاله: 1,384 تعداد دریافت فایل اصل مقاله: 1,788 |
||