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| Paper IPM / M / 8553 |
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| Abstract: | |
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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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