Zero-divisor graphs of non-commutative rings

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In a manner analogous to the commutative case, the zero-divisor graph of a non-commutative ring R can be defined as the directed graph Γ (R) that its vertices are all non-zero zero-divisors of R in which for any two distinct vertices x and y, x → y is an edge if and only if xy = 0. We investigate the interplay between the ring-theoretic properties of R and the graph-theoretic properties of Γ (R). In this paper it is shown that, with finitely many exceptions, if R is a ring and S is a finite semisimple ring which is not a field and Γ (R) ≃ Γ (S), then R ≃ S. For any finite field F and each integer n ≥ 2, we prove that if R is a ring and Γ (R) ≃ Γ (Mn), then R ≃ Mnn. Redmond defined the simple undirected graph Γ̄ (R) obtaining by deleting all directions on the edges in Γ (R). We classify all ring R whose Γ̄ (R) is a complete graph, a bipartite graph or a tree. © 2005 Published by Elsevier Inc.




Akbari, S., & Mohammadian, A. (2006). Zero-divisor graphs of non-commutative rings. Journal of Algebra, 296(2), 462–479.

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