Concerning real-closed ideals in [formula omitted] and SV-frames

Abstract

Let RL be the ring of continuous real-valued functions on a completely regular frame L. We study the class of prime ideals P of the ring RL determined by the condition: RL/P is a real-closed ring. We give some necessary and sufficient frame-theoretic conditions for an ordered integral domain of the form RL/P to be a real-closed ring, when P is a prime z-ideal (d-ideal) of RL. A completely regular frame L is called an SV-frame if for every prime ideal P of RL, the ordered integral domain RL/P is a real-closed ring. It is shown that every C⁎-quotient of an SV-frame is an SV-frame. We also show that open quotients ↓c in an SV-frame are SV-frames for all cozero elements c. Larson [22] has given a topological characterization of compact SV-spaces. By extending this characterization to frames, we show that the compactness limitation can really be relaxed, even in spaces, and so strengthen Larson's result.