When certain prime ideals in rings of continuous functions are minimal or maximal.
type: Item
instance of: scholarly article
title: When certain prime ideals in rings of continuous functions are minimal or maximal.
zbMATH Open document ID: 1325.06010
author: Themba Dube
author: Jissy Nsonde Nsayi
published in: Topology and its Applications
publication date: 2015-08-19T00:00:00Z
review text: Let \(A\) be a commutative ring with identity. The symbols \(\mathrm{Spec}(A)\), \(\mathrm{Max}(A)\), \(\mathrm{Min}(A)\), \(\mathrm{Spec}_d(A)\) and \(\mathrm{Spec}_z(A)\) denote the set of prime ideals, maximal prime ideals, minimal prime ideals, prime \(d\)-ideals and prime \(z\)-ideals of \(A\), respectively. The main goal of this paper is to characterize frames \(L\) such that, for each of the following conditions, the ring \(\mathcal RL\) of continuous real functions on \(L\) satisfies that condition: (dMin) \(\mathrm{Spec}_d(A)\subseteq\mathrm{Min}(A)\); (dMax) \(\mathrm{Spec}_d(A)\subseteq\mathrm{Max}(A)\); (dMM) \(\mathrm{Spec}_d(A)\subseteq\mathrm{Min}(A)\cup\mathrm{Max}(A)\); (zMin) \(\mathrm{Spec}_z(A)\subseteq\mathrm{Min}(A)\); (zMax) \(\mathrm{Spec}_z(A)\subseteq\mathrm{Max}(A)\); (zMM) \(\mathrm{Spec}_z(A)\subseteq\mathrm{Min}(A)\cup\mathrm{Max}(A)\). First, it is observed that in the class of rings with no nonzero nilpotent elements (i.e. reduced rings), (zMin), (zMax) and (dMax) are all equivalent, and are equivalent to the property of being von Neumann regular. This implies that \(\mathcal RL\) satisfies any of these three conditions if and only if \(L\) is a \(P\)-frame, a generalization of the classical notion of a \(P\)-space. Then it is proved that \(\mathcal RL\) satisfies (dMin) (resp. (dMM), resp. (zMM)) precisely if \(L\) is \textit{cozero complemented} (resp. \textit{quasi cozero complemented}, resp. a \textit{quasi \(P\)-frame}). In the last part of the article the authors examine how far the theory of quasi \(P\)-frames parallels that of quasi \(P\)-spaces as defined by \textit{M. Henriksen} et al. [Commentat. Math. Univ. Carol. 44, No. 2, 261-294 (2003; Zbl 1098.54013)].
reviewed by: Jorge Picado
Mathematics Subject Classification ID: 06D22
Mathematics Subject Classification ID: 13A15
Mathematics Subject Classification ID: 54G10
Mathematics Subject Classification ID: 54C30
Mathematics Subject Classification ID: 54G15
Mathematics Subject Classification ID: 54C40
zbMATH DE Number: 6473608
zbMATH Keywords: \(P\)-frames
zbMATH Keywords: quasi \(P\)-frames
zbMATH Keywords: almost \(P\)-frames
zbMATH Keywords: quasi cozero complemented frames
zbMATH Keywords: quasi \(m\)-spaces
zbMATH Keywords: maximal prime ideals
zbMATH Keywords: minimal prime ideals
zbMATH Keywords: rings of continuous real functions
MaRDI profile type: Publication
full work available at URL: j.topol.2015.05.073
OpenAlex ID: W923563142
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DOI: 10.1016/J.TOPOL.2015.05.073
description: scientific article; zbMATH DE number 6473608