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Paper   IPM / M / 7838
School of Mathematics
  Title:   Zero-Divisor graphs of non-commutative rings
1.  S. Akbari
2.  A. Mohammadian
  Status:   Published
  Journal: J. Algebra
  Vol.:  296
  Year:  2005
  Pages:   462-479
  Supported by:  IPM
    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 \mathnormalΓ(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. We investigate the interplay between the ring-theoretic properties of R and the graph-theoretic properties of \mathnormalΓ(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 \mathnormalΓ(R) ≅ \mathnormalΓ (S), then RS. For any finite field F and each integer n\geqslant2, we prove that if R is a ring and \mathnormalΓ(R) ≅ \mathnormalΓ(Mn(F)), then RMn(F). Redmond defined the simple undirected graph \mathnormalΓ(R) obtaining by deleting all directions on the edges in \mathnormalΓ(R). We classify all ring R whose \mathnormalΓ(R) is a complete graph, a bipartite graph or a tree.

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