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

Shane P. Redmond

• 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.