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On zero-divisor graphs of finite rings

Akbari, S ; Sharif University of Technology | 2007

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  1. Type of Document: Article
  2. DOI: 10.1016/j.jalgebra.2007.02.051
  3. Publisher: 2007
  4. Abstract:
  5. The zero-divisor graph of a ring R is 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 x y = 0. Recently, it has been shown that for any finite ring R, Γ (R) has an even number of edges. Here we give a simple proof for this result. In this paper we investigate some properties of zero-divisor graphs of matrix rings and group rings. Among other results, we prove that for any two finite commutative rings R, S with identity and n, m ≥ 2, if Γ (Mn (R)) ≃ Γ (Mm (S)), then n = m, | R | = | S |, and Γ (R) ≃ Γ (S). © 2007 Elsevier Inc. All rights reserved
  6. Keywords:
  7. Eulerian graph ; Group ring ; Matrix ring ; Zero-divisor graph
  8. Source: Journal of Algebra ; Volume 314, Issue 1 , 2007 , Pages 168-184 ; 00218693 (ISSN)
  9. URL: https://www.sciencedirect.com/science/article/pii/S0021869307001226