Unions of minimal prime ideals in rings of continuous functions on compact spaces
type: Item
instance of: scholarly article
title: Unions of minimal prime ideals in rings of continuous functions on compact spaces
zbMATH Open document ID: 1200.54009
author: Bikram Banerjee
author: Swapan Kumar Ghosh
author: M. Henriksen
published in: Algebra Universalis
publication date: 2010-08-13T00:00:00Z
review text: In the ring \(C(X)\) of real-valued continuous functions on a Tychonoff space \(X\), every prime ideal is contained in a unique maximal ideal. Using the language of Gillman, Jerison, and Henriksen, the authors study the union of all the minimal prime ideals contained in the maximal ideal \(M_p = \{f \in C(X) : f(p) = 0\}\) for some \(p \in X\). A motivation for the study is given in the abstract of the paper. A space \(X\) is called a UMP-space if every maximal ideal of \(C(X)\) is the union of the minimal prime ideals contained in it. Every P-space (in which every prime ideal of \(C(X)\) is maximal) is a UMP-space, which in turn is an almost-P-space (in which every element of a minimal prime ideal of \(C(x)\) is a zero-divisor [\textit{A. I. Veksler}, Sov. Math., Dokl. 14, 1445--1450 (1973); translation from Dokl. Akad. Nauk SSSR 212, 789--792 (1973; Zbl 0291.54046), \textit{R. Levy}, Can. J. Math. 29, 284--288 (1977; Zbl 0342.54032)]. The almost-P-space \(\beta \omega - \omega\) is not a UMP-space. If \(D\) is an uncountable discrete space, \(\alpha D = D \cup \{d\}\) is the one-point compactification of \(D\), then \(\alpha D\) is a UMP-space. A Cantor set is not UMP, being metrizable but not discrete. The paper gives an algebraic characterization of a UMP-space when \(X\) is compact.
Mathematics Subject Classification ID: 54C40
Mathematics Subject Classification ID: 13J25
zbMATH DE Number: 5770086
zbMATH Keywords: UMP space
zbMATH Keywords: UMP point
zbMATH Keywords: almost P-space
zbMATH Keywords: minimal prime ideal
zbMATH Keywords: round set
zbMATH Keywords: nearly round set
zbMATH Keywords: almost round set
reviewed by: Pao-sheng Hsu
MaRDI profile type: Publication
full work available at URL: s00012-010-0051-x
OpenAlex ID: W1986687231
cites work: Q3417819
cites work: Q4092470
cites work: The Space of Minimal Prime Ideals of a Commutative Ring
cites work: Almost-P-Spaces
cites work: Round \(z\)-filters and round subsets of \(\beta X\)
cites work: Q4042338
DOI: 10.1007/S00012-010-0051-X
description: scientific article; zbMATH DE number 5770086