Rings in which every prime ideal is contained in a unique maximal right ideal
type: Item
instance of: scholarly article
title: Rings in which every prime ideal is contained in a unique maximal right ideal
zbMATH Open document ID: 0774.16001
DOI: 10.1016/0022-4049(92)90096-X
author: Shu-Hao Sun
published in: Journal of Pure and Applied Algebra
publication date: 1992-09-27T00:00:00Z
review text: In order to extend to rings the Gelfand duality between compact spaces and commutative \(C^*\) algebras, \textit{C. J. Mulvey} [J. Algebra 56, 499--505 (1979; Zbl 0498.16035)] introduced the following condition: Let \(R\) be a ring such that for any distinct maximal right ideals \(M\), \(M'\), there exist \(a\notin M\), \(a'\notin M'\) in \(R\) with \(aRa'=0\). Then \(R\) is called a Gelfand ring. It follows that every prime ideal in such a ring is contained in a unique maximal right ideal. In this paper, rings with this last property and related properties of the space \(\text{spec}_ r R\) of all prime right ideals and \(\text{spec }R\) are considered. The second section of the paper is concerned with set theoretical axioms; in particular it is shown that under the assumption of the prime ideal theorem, and not assuming the axiom of choice, the lattice of right ideals of \(R\) is normal if and only if \(R\) satisfies the maximal ideal theorem and is Gelfand.
Mathematics Subject Classification ID: 16D25
Mathematics Subject Classification ID: 16N60
Mathematics Subject Classification ID: 16D90
zbMATH DE Number: 60684
zbMATH Keywords: Gelfand duality
zbMATH Keywords: maximal right ideals
zbMATH Keywords: Gelfand ring
zbMATH Keywords: prime right ideals
zbMATH Keywords: prime ideal theorem
zbMATH Keywords: axiom of choice
zbMATH Keywords: lattice of right ideals
zbMATH Keywords: maximal ideal theorem
reviewed by: H. H. Brungs
MaRDI profile type: Publication
cites work: A Sheaf Representation for Modules with Applications to Gelfand Rings
cites work: Algebra in a localic topos with applications to ring theory
cites work: Commutative Rings in Which Every Prime Ideal is Contained in a Unique Maximal Ideal
cites work: A generalisation of Gel'fand duality
cites work: Q3752439
cites work: Erratum to: Reticulated rings
cites work: Noncommutative rings in which every prime ideal is contained in a unique maximal ideal
cites work: On separation lemmas
full work available at URL: 0022-4049(92)90096-x
OpenAlex ID: W1987699381
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description: scientific article; zbMATH DE number 60684