Integral extensions on rings of continuous functions
type: Item
instance of: scholarly article
title: Integral extensions on rings of continuous functions
zbMATH Open document ID: 1182.54022
author: M. A. Mulero
author: Jesús M. Domínguez
published in: Topology and its Applications
publication date: 2009-11-05T00:00:00Z
review text: Observing that a realcompact space \(X\) is determined by the algebra \(C(X)\) of all real-valued continuous functions on \(X\) and that continuous maps between such spaces are in 1-1 correspondence with morphisms between their algebras of continuous functions, one or both of the authors have investigated in a series of papers [e.g., \textit{M. A. Mulero}, Fundam. Math. 149, No.~1, 55--66 (1996; Zbl 0840.54020); the authors, Topology Appl. 137, No.~1--3, 115--124 (2004; Zbl 1045.54007)] the algebraic properties of rings of continuous function in relation to the topological properties of mappings between topological spaces. In this paper, a surjective continuous map \(\pi : X \to Y\) between Tychonoff spaces induces an injective morphism \(C(Y) \to C(X)\) by composition: \(g \mapsto g\circ\pi\), which turns \(C(X)\) into a \(C(Y)\)-algebra. Here, \(C(Y)\) considered as a subring of \(C(X)\), the extension \(C(Y)\subseteq C(X)\) is integral if every element in \(C(X)\) satisfies a monic polynomial equation over \(C(Y)\). The main result is that if \(X\) is a compact subset of \(\mathbb R^n\), the extension \(C(Y)\subseteq C(X)\) is integral if and only if \(X\) decomposes into a finite union of closed subsets such that \(\pi\) is injective on each. In the absence of compactness, if the extension \(C(Y)\subseteq C(X)\) is integral, then the supremum of the cardinalities of the fibers of \(\pi\) is finite. The Cantor ternary set \(\mathbf {C}\) and a map \(\pi : \mathbf {C} \to I = [0, 1]\) are given and shown as an example in which a continuous onto map between compact Hausdorff spaces has an upper bound for the cardinality of the fibers of \(\pi\) but \(C(I)\subseteq C(\mathbf {C})\) is not integral.
Mathematics Subject Classification ID: 54C40
Mathematics Subject Classification ID: 13B21
Mathematics Subject Classification ID: 54D35
zbMATH DE Number: 5625496
zbMATH Keywords: rings of continuous functions
zbMATH Keywords: integral extension
zbMATH Keywords: finitely generated extension
zbMATH Keywords: monic polynomial
zbMATH Keywords: integral closure
reviewed by: Pao-sheng Hsu
MaRDI profile type: Publication
full work available at URL: j.topol.2009.01.019
OpenAlex ID: W1992208681
cites work: Q5563439
cites work: Q4107197
cites work: On covering spaces and Galois extensions
cites work: Finite homomorphisms on rings of continuous functions.
cites work: Stone-Čech compactification of a product
cites work: Q5627356
cites work: Q4868603
cites work: Q4222596
DOI: 10.1016/J.TOPOL.2009.01.019
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description: scientific article; zbMATH DE number 5625496