COMPACTNESS OF A SUBSPACE OF THE ZARISKI TOPOLOGY ON SPEC(D)

• Journal title : Honam Mathematical Journal
• Volume 33, Issue 3,  2011, pp.419-424
• Publisher : The Honam Mathematical Society
• DOI : 10.5831/HMJ.2011.33.3.419
Title & Authors
COMPACTNESS OF A SUBSPACE OF THE ZARISKI TOPOLOGY ON SPEC(D)
Chang, Gyu-Whan;

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 $\small{I{\nsubseteq}P}$ for all $\small{P{\in}X}$, there exists a finitely generated idea $\small{J{\subseteq}I}$ such that $\small{J{\nsubseteq}P}$ for all $\small{P{\in}X}$. We also prove that if D = $\small{{\cap}_{P{\in}X}D_P}$ and if * is the star-operation on D induced by X, then X is compact if and only if * $\small{_f}$-Max(D) $\small{{\subseteq}}$X. As a corollary, we have that t-Max(D) is compact and that $\small{{\mathcal{P}}}$(D) = {P$\small{{\in}}$ Spec(D)$\small{|}$P is minimal over (a : b) for some a, b$\small{{\in}}$D} is compact if and only if t-Max(D) $\small{{\subseteq}\;{\mathcal{P}}}$(D).
Keywords
Zariski topology;subspace topology;compactness;* $\small{_f}$-Max(D);$\small{\mathcal{P}}$(D);
Language
English
Cited by
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