The zero-divisor graph of a reduced ring

Abstract

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 ) .