Compactness of a Subspace of the Zariski Topology on Spec(d)

Abstract

Abstract. Let D be an integral domain, Spec(D) the set of prime ideals of D, and X a subspace of the Zariski topology on Spec(D). We show that X is compact if and only if given any ideal I of D with I P for all P ∈ X, there exists a finitely generated ideal J ⊆ I such that J P for all P ∈ X. We also prove that if D = ∩P ∈X DP and if * is the star-operation on D induced by X, then X is compact if and only if * f -Max(D) ⊆ X. As a corollary, we have that t-Max(D) is compact and that P(D) = {P ∈ Spec(D)|P is minimal over (a : b) for some a, b ∈ D} is compact if and only if t-Max(D) ⊆ P(D).