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REAL IDEALS IN POINTFREE RINGS OF CONTINUOUS FUNCTIONS

Part of: Boolean algebras (Boolean rings) Modules, bimodules and ideals Distributive lattices

Published online by Cambridge University Press:  06 December 2010

THEMBA DUBE
THEMBA DUBE*
Affiliation:
Department of Mathematical Sciences, University of South Africa, PO Box 392, 0003 Unisa, South Africa (email: dubeta@unisa.ac.za)
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Abstract

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Real ideals of the ring ℜL of real-valued continuous functions on a completely regular frame L are characterized in terms of cozero elements, in the manner of the classical case of the rings C(X). As an application, we show that L is realcompact if and only if every free maximal ideal of ℜL is hyper-real—which is the precise translation of how Hewitt defined realcompact spaces, albeit under a different appellation. We also obtain a frame version of Mrówka’s theorem that characterizes realcompact spaces.

Keywords

framering of real-valued continuous functions on a frameidealreal idealhyper-real idealrealcompact

MSC classification

Secondary: 06D22: Frames, locales 16D25: Ideals
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 83 , Issue 2 , April 2011 , pp. 338 - 352
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2010

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