Finite homomorphisms on rings of continuous functions.
type: Item
instance of: scholarly article
title: Finite homomorphisms on rings of continuous functions.
zbMATH Open document ID: 1045.54007
DOI: 10.1016/S0166-8641(03)00203-7
author: M. A. Mulero
author: Jesús M. Domínguez
published in: Topology and its Applications
publication date: 2004-03-14T00:00:00Z
review text: Given the rings \(A\), \(B\) and the homomorphism \(h\) from \(A\) into \(B\), we say that \(h\) is finite in case \(B\) is a finitely generated \(A\)-module. Let \(X\), \(Y\) be compact Hausdorff spaces and \(\pi\) be a continuous mapping from \(X\) into \(Y\). The authors' main result is that the induced homomorphism \(h\) from the ring \(C(X)\) of continuous real valued functions on \(X\) into the ring \(C(Y)\) is finite if, and only if, \(\pi\) is locally injective. They extend this result to the case where \(X\), \(Y\) are realcompact and \(Y\) is normal. Then \(h\) is finite if, and only if, \(\pi\) is a closed mapping and \(X\) can be covered by a finite number of cozero sets on each of which \(\pi\) is injective. Some results of \textit{G. D. Faulkner} [Proc. Am. Math. Soc. 108, No. 2, 541--546 (1990; Zbl 0679.54023)] are obtained as particular cases of their results.
reviewed by: James V. Whittaker
Mathematics Subject Classification ID: 54C40
Mathematics Subject Classification ID: 13B21
Mathematics Subject Classification ID: 54D35
Mathematics Subject Classification ID: 13E15
zbMATH DE Number: 2056847
zbMATH Keywords: Rings of continuous functions
zbMATH Keywords: Compactifications
zbMATH Keywords: Finite homomorphisms
zbMATH Keywords: Integer dependence
MaRDI profile type: Publication
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full work available at URL: s0166-8641(03)00203-7
OpenAlex ID: W1994375643
description: scientific article; zbMATH DE number 2056847