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Paper   IPM / M / 8538
School of Mathematics
  Title:   On zero-divisor graphs of finite rings
1.  S. Akbari
2.  A. Mohammadian
  Status:   Published
  Journal: J. Algebra
  Vol.:  314
  Year:  2007
  Pages:   168-184
  Supported by:  IPM
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, xy is an edge if and only if xy = 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\geqslant 2, if Γ(Mn(R)) ≅ Γ(Mm(S)), then n = m, |R| = |S|, and Γ(R) ≅ Γ(S)

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