Ring ideals and the Stone-Čech compactification in pointfree topology
type: Item
instance of: scholarly article
title: Ring ideals and the Stone-Čech compactification in pointfree topology
zbMATH Open document ID: 1206.06005
author: M. Sioen
author: Bernhard Banaschewski
published in: Journal of Pure and Applied Algebra
publication date: 2010-08-27T00:00:00Z
review text: There are several ways of constructing the compact completely regular coreflection, \(\beta L\), of a completely regular frame \(L\). The original construction, which is lattice-theoretic and realizes \(\beta L\) as the frame of completely regular ideals of \(L\), goes back to \textit{B. Banaschewski} and \textit{C. J. Mulvey} [Houston J. Math. 6, 301--312 (1980; Zbl 0473.54026)]. There have subsequently been other constructions in terms of \(\ell\)-ideals of the \(f\)-ring \(\mathbb RL\) of continuous real-valued functions on \(L\) [see \textit{B. Banaschewski}, The real numbers in pointfree topology. Textos de Matemática. Série B. 12. Coimbra: Universidade de Coimbra, Departamento de Matemática (1997; Zbl 0891.54009)]. In the paper under review, \(\beta L\) is realized in terms of mere ring ideals of \(\mathbb RL\). Namely, it is the frame of Jacobson radical ideals of \(\mathbb RL\). Analogously, the compact zero-dimensional coreflection of \(L\) is described in terms of mere ring ideals of the ring \(\mathbb ZL\) of continuous integer-valued functions on \(L\). As the authors point out, the constructions above can be obtained by putting together several results in some papers by the first author. However, the direct proofs in the article accentuate the clarity of the results with great elegance.
reviewed by: Themba Dube
Mathematics Subject Classification ID: 06D22
Mathematics Subject Classification ID: 06F25
Mathematics Subject Classification ID: 54C30
Mathematics Subject Classification ID: 54D30
zbMATH DE Number: 5775710
zbMATH Keywords: frame
zbMATH Keywords: Stone-Čech compactification of a frame
zbMATH Keywords: pointfree ring of continuous functions
zbMATH Keywords: Jacobson radical ideals
MaRDI profile type: Publication
full work available at URL: j.jpaa.2010.02.019
OpenAlex ID: W2059205268
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DOI: 10.1016/J.JPAA.2010.02.019
description: scientific article; zbMATH DE number 5775710