Ideals associated with realcompactness in pointfree function rings

Published in: Quaestiones Mathematicae
Volume 38, issue 6, 2015, pages: 885–899
DOI: 10.2989/16073606.2015.1015648
Author(s): Themba DubeDepartment of Mathematical Sciences, South Africa


Let RL denote the ring of continuous real-valued functions on a com- pletely regular frame L. The support of an αRL 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 satisfies 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 υLL is a one-point extension of L.

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