Abstract

Let P be a prime ideal of the ring of continuous real-valued functions on a completely regular frame L, i.e., L. We study many new results about the residue class domains L/P with an emphasis on determining when the ordered L/P is a valuation domain (i.e., when given any two non-zero elements of L/P , one divides the other). A prime ideal P of L is called a valuation prime ideal if L/P is a valuation domain. A frame L is called an SV-frame if every prime ideal of L is a valuation prime ideal. We introduce and study two new generalizations of the SV-frames. The first is that of a quasi SV-frame in which every real maximal ideal of L that is not a minimal prime ideal contains a non-maximal prime ideal P such that L/P is a valuation domain. In the second, we define a frame L to be an almost SV-frame if every maximal ideal of L contains a minimal valuation prime ideal. A point I ∈ Pt(βL) is called a special βF -point if O^I = {δ ∈L : coz δ ∈ I} is a valuation prime ideal of L. It is shown that I is a special βF -point if and only if the pseudo-prime ideals of L containing O^I that are not primary form a chain under set inclusion.

摘要

令P是完全正则框架L上的连续实值函数环的素理想，即L。我们研究了许多关于残基类域L/P的新结果，重点是确定有序L/P何时是估值域（即，当给定L/P的任何两个非零元素时，一个除以另一个）。如果L/P是估值域，则L的素理想P称为估值素理想。如果L的每个素理想，则框架L称为SV框架是估值素理想。我们介绍并研究了SV框架的两个新的推广。第一个是准SV框架，其中L的每个不是最小素理想的实最大理想都包含一个非最大素理想P，使得L/P是一个估值域。其次，如果L的每个最大理想都包含一个最小估值素理想，我们将框架L定义为几乎SV框架。如果O ^I = { δ ∈ L : coz _ _ _δ ∈ I}是L的一个估值素理想。证明了I是一个特殊的βF点当且仅当包含O ^I的L的伪素理想在集合包含下形成一条链。