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Paper   IPM / M / 8553
School of Mathematics
  Title:   The zero-divisor graph of a reduced ring
  Author(s):  K. Samei
  Status:   Published
  Journal: J. Pure Appl. Algebra
  Vol.:  209
  Year:  2007
  Pages:   813-821
  Supported by:  IPM
In this paper the zero-divisor graph Γ(R) of a commutative reduced ring R is studied. We associate the ring properties of R, the graph properties of Γ(R) and the topological properties of Spec(R). Cycles in Γ(R) are investigated and an algebraic and a topological characterization is given for the graph Γ(R) to be triangulated or hypertriangulated. We show that the clique number of Γ(R), the cellularity of Spec(R) and the Goldie dimension of R coincide. We prove that when R has the annihilator condition and 2 ∉ Z(R);Γ(R) is complemented if and only if Min(R) is compact. In a semiprimitive Gelfand ring, it turns out that the dominating number of Γ(R) is between the density and the weight of Spec(R). We show that Γ(R) is not triangulated and the set of centers of Γ(R) is a dominating set if and only if the set of isolated points of Spec(R) is dense in Spec(R).

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