Ideals associated with realcompactness in pointfree function rings

Abstract

Let RL denote the ring of continuous real-valued functions on a completely regular frame L. The support of an α ∈ RL is the closed quotient ↑(coz α ). We show that if supports are coz-quotients in L, then the set of functions with realcompact support is an ideal. If L satises the stronger condition that supports are C-quotients, then this ideal is the intersection of pure parts of the free maximal ideals of RL. The set of functions whose cozeroes are realcompact is always an ideal, which is free if and only if L is locally realcompact if and only if L is (isomorphic to) an open quotient of υL. Further, this ideal is prime if and only if it is a free real maximal ideal if and only if υL → L is a one-point extension of L.

Keywords: Frame, ring of continuous real functions on a frame, support, realcompact support, locally realcompact frame,  Lindelöf support