Extending ring topologies
type: Item
instance of: scholarly article
title: Extending ring topologies
zbMATH Open document ID: 0968.16020
author: Dieter Remus
author: Horst Szambien
author: W. W. Comfort
published in: Journal of Algebra
publication date: 2001-06-18T00:00:00Z
review text: The authors study the question of strengthening totally bounded (= precompact and Hausdorff) ring topologies on rings. The main tool is the concept of the Bohr compactification of a topological ring. It is established for a given discrete ring \(R\) an antiisomorphism between the lattice of precompact topologies on \(R\) and the lattice of closed ideals of its Bohr compactification \(bR\). There are given suffient conditions for the existence for a compact ring \((R,{\mathfrak T})\) of pseudocompact ring topologies extending \(\mathfrak T\): Theorem. Let \((R,{\mathfrak T})\) be a compact ring for which \(\alpha:=w(R,{\mathfrak T})\) satisfies \(\text{cf}(\alpha)>\omega\). Assume that either (i) \(w(R/J(R))=\alpha\), or (ii) \(R\) is semisimple. Then there exist exactly \(2^{2^{|R|}}\)-many pseudocompact topologies \(\mathfrak V\) on \(R\) such that \({\mathfrak V}\supseteq{\mathfrak T}\). A topological ring \((R,{\mathfrak T})\) is said to be a van der Waerden ring provided each ring homomorphism \(\alpha\) in any compact ring \((R',{\mathfrak T}')\) is continuous. It is given in the paper a complete characterization of semisimple in the sense of Jacobson compact van der Waerden rings. It is proved that compact Noetherian rings are van der Waerden rings. It is also proved that a matrix ring over a van der Waerden ring with identity is a van der Waerden ring. In the last part of the paper is calculated the Bohr compactification of \(\mathbb{Z}\) and are given different examples of van der Waerden rings. The authors make interesting remarks concerning the analogy between the results of the paper under review and the results concerning Abelian groups contained in earlier papers. They note (page 33), for example, that in contrast with the case of Abelian groups, there exist van der Waerden rings having non-closed ideals.
Mathematics Subject Classification ID: 16W80
Mathematics Subject Classification ID: 54A10
zbMATH DE Number: 1539373
zbMATH Keywords: semisimple rings
zbMATH Keywords: Jacobson radical
zbMATH Keywords: radical rings
zbMATH Keywords: Kaplansky representations
zbMATH Keywords: topological rings
zbMATH Keywords: compact rings
zbMATH Keywords: totally bounded rings
zbMATH Keywords: precompact rings
zbMATH Keywords: pseudocompact rings
zbMATH Keywords: Weil completions
zbMATH Keywords: Bohr compactifications
zbMATH Keywords: van der Waerden rings
zbMATH Keywords: \(p\)-adic integers
zbMATH Keywords: \(p\)-adic topology
zbMATH Keywords: lattices of topologies
reviewed by: Mihail I. Ursul
MaRDI profile type: Publication
full work available at URL: jabr.2000.8382
OpenAlex ID: W1977272961
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DOI: 10.1006/JABR.2000.8382
description: scientific article; zbMATH DE number 1539373