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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 $I{\nsubseteq}P$ for all $P{\in}X$, there exists a finitely generated idea $J{\subseteq}I$ such that $J{\nsubseteq}P$ for all $P{\in}X$. We also prove that if D = ${\cap}_{P{\in}X}D_P$ and if * is the star-operation on D induced by X, then X is compact if and only if * $_f$-Max(D) ${\subseteq}$X. As a corollary, we have that t-Max(D) is compact and that ${\mathcal{P}}$(D) = {P${\in}$ Spec(D)$|$P is minimal over (a : b) for some a, b${\in}$D} is compact if and only if t-Max(D) ${\subseteq}\;{\mathcal{P}}$(D).
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참고문헌 (8)
참고문헌 신청
D.D. Anderson / 1988 / Star-operations induced by overrings / Comm. Algebra 16:2535~2553
M.F. Atiyah / 1968 / Introduction to Commutative Algebra / Addison-Wesley
R. Gilmer / 1972 / Multiplicative Ideal Theory / Marcel Dekker
E. Houston / 1989 / On t-invertibility II / Comm. Algebra 17:1955~1969
J. Mott / 1981 / On Prufer v-multiplication domain / Manuscripta Math 35:1~26
M.F. Atiyah / 1968 / Introduction to Commutative Algebra / Addison-Wesley
R. Gilmer / 1972 / Multiplicative Ideal Theory / Marcel Dekker
E. Houston / 1989 / On t-invertibility II / Comm. Algebra 17:1955~1969
J. Mott / 1981 / On Prufer v-multiplication domain / Manuscripta Math 35:1~26
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