Structure in the Zero-Divisor Graph of a Non-Commutative Ring

• Autores: Shane P. Redmond
• Localización: Houston journal of mathematics, ISSN 0362-1588, Vol. 30, Nº 2, 2004, págs. 345-355
• Idioma: inglés
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• Resumen

  • In a manner analogous to the commutative case, the zero-divisor graph of a noncommutative ring R can be defined as the directed graph G. It has been shown that G is not a tournament if R is a finite ring with no nontrivial nilpotent elements and the graph has more than one vertex. This result is generalized to an arbitrary ring. This article also shows that G cannot be a network for a finite ring R. These results are used to determine which directed graphs on 1, 2, or 3 vertices can be realized as G. Finally, it is shown that for a finite ring R, G has an even number of directed edges.