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Zero-divisor graphs of non-commutative rings

Akbari, S ; Sharif University of Technology | 2006

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  1. Type of Document: Article
  2. DOI: 10.1016/j.jalgebra.2005.07.007
  3. Publisher: 2006
  4. Abstract:
  5. 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
  6. Keywords:
  7. Directed graph ; Zero-divisor ; Non-commutative ring ; Matrix ring
  8. Source: Journal of Algebra ; Volume 296, Issue 2 , 2006 , Pages 462-479 ; 00218693 (ISSN)
  9. URL: https://www.sciencedirect.com/science/article/pii/S0021869305004023