\(C(X)\) versus its functionally countable subalgebra
type: Item
instance of: scholarly article
title: \(C(X)\) versus its functionally countable subalgebra
zbMATH Open document ID: 1416.54006
author: Mostafa Ghadermazi
author: M. Namdari
author: O. A. S. Karamzadeh
published in: Bulletin of the Iranian Mathematical Society
publication date: 2019-03-22T00:00:00Z
review text: In the spirit of the study of rings of continuous functions, the authors study the interplay between some topological properties of zero-dimensional spaces $X$ and algebraic properties of the rings $C_c(X)$ of continuous real-valued functions on $X$ having countable image. They show, among other things, that two zero-dimensional spaces $X$ and $Y$ are homeomorphic if and only if $C_c(X)$ and $C_c(Y)$ are isomorphic. They characterize spaces $X$ for which every function in $C_c(X)$ actually has finite image. These are precisely the spaces $X$ for which $C_c(X)$ is von Neumann regular and each locally finite family of clopen subsets of $X$ is finite. A curious observation is that, whereas the socle of $C(X)$ is never a prime ideal in $C(X)$, there are instances when it is a prime ideal in $C_c(X)$. The authors actually characterize the spaces for which this is the case.
reviewed by: Themba Dube
Mathematics Subject Classification ID: 54C40
Mathematics Subject Classification ID: 13C11
zbMATH DE Number: 7040768
zbMATH Keywords: rings of continuous functions
zbMATH Keywords: functionally countable subring
zbMATH Keywords: socle
zbMATH Keywords: ideal
zbMATH Keywords: regular ring
zbMATH Keywords: \(CP\)-space
MaRDI profile type: Publication
full work available at URL: s41980-018-0124-8
OpenAlex ID: W3002825945
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DOI: 10.1007/S41980-018-0124-8
description: scientific article; zbMATH DE number 7040768