Quasi SV-Frames and their properties

Abstract

Let P be a prime ideal of the ring of continuous real-valued functions on a completely regular frame L, i.e., RL. We study many new results about the residue class domains RL=P with an emphasis on determining when the ordered RL=P is a valuation domain (i.e., when given any two non-zero  elements of RL=P, one divides the other). A prime ideal P of RL is called a valuation prime ideal if RL=P is a valuation domain. A frame L is called an  SV-frame if every prime ideal of RL is a valuation prime ideal. We introduce and study two new generalizations of the SV-frames. The rst is that of a  quasi SV-frame in which every real maximal ideal of RL that is not a minimal prime ideal contains a non-maximal prime ideal P such that RL=P is a  valuation domain. In the second, we dene a frame L to be an almost SV-frame if every maximal ideal of RL contains a minimal valuation prime ideal.  A point I 2 Pt(βL) is called a special βF-point if O^I = {δ∈RL : coz δ∈I is a valuation prime ideal of RL. It is shown that I is a special F-point if and only if the  pseudo-prime ideals of RL containing OI that are not primary form a chain under set inclusion.