ON THE REFINEMENT OF THE UNIT AND UNITARY CAYLEY GRAPHS OF RINGS | ||
| Journal of Algebraic Systems | ||
| مقاله 4، دوره 7، شماره 1، آذر 2019، صفحه 51-68 اصل مقاله (305.19 K) | ||
| نوع مقاله: Original Manuscript | ||
| شناسه دیجیتال (DOI): 10.22044/jas.2018.6939.1340 | ||
| نویسندگان | ||
| M. Rezagholibeigi؛ A. R. Naghipour* | ||
| Department of Mathematical Sciences, Shahrekord University, P.O.Box 115, Shahrekord, Iran. | ||
| چکیده | ||
| Let $R$ be a ring (not necessarily commutative) with nonzero identity. We define $\Gamma(R)$ to be the graph with vertex set $R$ in which two distinct vertices $x$ and $y$ are adjacent if and only if there exist unit elements $u,v$ of $R$ such that $x+uyv$ is a unit of $R$. In this paper, basic properties of $\Gamma(R)$ are studied. We investigate connectivity and the girth of $\Gamma(R)$, where $R$ is a left Artinian ring. We also determine when the graph $\Gamma(R)$ is a cycle graph. We prove that if $\Gamma(R)\cong\Gamma(M_{n}(F))$ then $R\cong M_{n}(F)$, where $R$ is a ring and $F$ is a finite field. We show that if $R$ is a finite commutative semisimple ring and $S$ is a commutative ring such that $\Gamma(R)\cong\Gamma(S)$, then $R\cong S$. Finally, we obtain the spectrum of $\Gamma(R)$, where $R$ is a finite commutative ring. | ||
| کلیدواژهها | ||
| Rings؛ Matrix rings؛ Jacobson radical؛ Unit graphs؛ Unitary Cayley graphs | ||
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آمار تعداد مشاهده مقاله: 737 تعداد دریافت فایل اصل مقاله: 808 |
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