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On zero-divisor graphs of quotient rings and complemented zero-divisor graphs | ||
| Journal of Algebra and Related Topics | ||
| مقاله 5، دوره 4، شماره 1، شهریور 2016، صفحه 39-50 اصل مقاله (312.35 K) | ||
| نوع مقاله: Research Paper | ||
| نویسندگان | ||
| P. Karimi Beiranvand* 1؛ R. Beyranvand2 | ||
| 1Islamic Azad university, Khorramabad Branch, Khorramabad | ||
| 2Lorestan University | ||
| چکیده | ||
| For an arbitrary ring $R$, the zero-divisor graph of $R$, denoted by $\Gamma (R)$, is an undirected simple graph that its vertices are all nonzero zero-divisors of $R$ in which any two vertices $x$ and $y$ are adjacent if and only if either $xy=0$ or $yx=0$. It is well-known that for any commutative ring $R$, $\Gamma (R) \cong \Gamma (T(R))$ where $T(R)$ is the (total) quotient ring of $R$. In this paper we extend this fact for certain noncommutative rings, for example, reduced rings, right (left) self-injective rings and one-sided Artinian rings. The necessary and sufficient conditions for two reduced right Goldie rings to have isomorphic zero-divisor graphs is given. Also, we extend some known results about the zero-divisor graphs from the commutative to noncommutative setting: in particular, complemented and uniquely complemented graphs. | ||
| کلیدواژهها | ||
| Quotient ring؛ zero-divisor graph؛ reduced ring؛ complemented graph | ||
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