A principal ideal theorem for compact sets of rank one valuation rings

作者：Olberding Bruce^*

来源：Journal of Algebra, 2017, 489: 399-426.

DOI：10.1016/j.jalgebra.2017.06.028

• Valuation ring
• Zariski-Riemann space
• Prefer domain
• Bezout domain

Let F be a field, and let Zar(F) be the space of valuation rings of F with respect to the Zariski topology. We prove that if X is a quasicompact set of rank one valuation rings in Zar(F) whose maximal ideals do not intersect to 0, then the intersection of the rings in X is an integral domain with quotient field F such that every finitely generated ideal is a principal ideal. To prove this result, we develop a duality between (a) quasicompact sets of rank one valuation rings whose maximal ideals do not intersect to 0, and (b) one-dimensional Prufer domains with nonzero Jacobson radical and quotient field F. The necessary restriction in all these cases to collections of valuation rings whose maximal ideals do not intersect to 0 is motivated by-settings in which the valuation rings considered all dominate a given local ring.

• 出版日期2017-11-1