Contracting the socle in rings of continuous functions
type: Item
instance of: scholarly article
title: Contracting the socle in rings of continuous functions
zbMATH Open document ID: 1200.06005
DOI: 10.4171/RSMUP/123-2
author: Themba Dube
published in: Rendiconti del Seminario Matematico della Università di Padova
publication date: 2010-09-07T00:00:00Z
full work available at URL: https://eudml.org/doc/243271
full work available at URL: http://rendiconti.math.unipd.it/volumes/vol123.php
review text: The socle of a ring is the ideal generated by the minimal ideals of the ring. This paper starts with the classical result that for a completely regular Hausdorff space \(X\), the socle of the ring \(C(X)\) is the ideal consisting of all functions that are zero everywhere except on a finite number of points. The goal of the paper is to study the socle of the ring \({\mathcal R}L\) of real continuous functions on a frame (locale) \(L\). The author starts by characterizing the socle of \({\mathcal R}L\): it is the ideal consisting of all functions whose cozero elements are finite joins of atoms of \(L\). The spatial result above follows as an easy corollary. It also follows that for any surjective frame homomorphism \(h:L\rightarrow M\), the induced ring homomorphism \({\mathcal R}h:{\mathcal R}L\rightarrow {\mathcal R}M\) extends the socle to an ideal contained in the socle (that is, \(({\mathcal R}h)[\mathrm{Soc}\, {\mathcal R}L]\subseteq \mathrm{Soc}\, {\mathcal R}M\)). Then it is proved that \(\mathrm{Soc}\, {\mathcal R}L\) is an exoteric ideal and that \({\mathcal R}h\) is exoteric for every nearly open frame homomorphism \(h\). Finally, a class of frame homomorphisms, called W-maps, is introduced. These maps characterize \(P\)-frames as those \(L\) for which every frame homomorphism with domain \(L\) is of this kind. Every dense W-map \(h:L\rightarrow M\) induces a ring homomorphism \({\mathcal R}h\) that contracts the socle of its codomain to an ideal contained in the socle of its domain (that is, \(({\mathcal R}h)^{-1}[\mathrm{Soc}\, {\mathcal R}M]\subseteq \mathrm{Soc} \, {\mathcal R}L\)). In case \(h\) is also surjective, \({\mathcal R}h\) contracts the socle of \({\mathcal R}M\) to the socle of \({\mathcal R}L\) (that is, \(({\mathcal R}h)^{-1}[\mathrm{Soc}\, {\mathcal R}M]= \mathrm{Soc} \, {\mathcal R}L\)).
reviewed by: Jorge Picado
Mathematics Subject Classification ID: 06D22
Mathematics Subject Classification ID: 13A15
Mathematics Subject Classification ID: 13J25
Mathematics Subject Classification ID: 54C30
Mathematics Subject Classification ID: 54C40
zbMATH DE Number: 5780243
zbMATH Keywords: ring of continuous functions
zbMATH Keywords: socle
zbMATH Keywords: annihilator
zbMATH Keywords: exoteric ring
zbMATH Keywords: frame homomorphism
zbMATH Keywords: nearly open frame homomorphism
zbMATH Keywords: cozero map
zbMATH Keywords: \(P\)-frame
zbMATH Keywords: W-map
zbMATH Keywords: Stone-Čech compactification
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 5780243