Radicals commuting with Cartesian products
type: Item
instance of: scholarly article
title: Radicals commuting with Cartesian products
zbMATH Open document ID: 0936.20045
author: A. L. S. Corner
author: Rüdiger Göbel
published in: Archiv der Mathematik
publication date: 2000-02-27T00:00:00Z
review text: Given an Abelian group \(G\), the group radical is defined by \(R_G(X)=\bigcap\{\text{Ker }\phi\mid\phi\colon X\to G\}\), for Abelian groups \(X\). This radical does not always commute with infinite direct products (for instance, when \(G=\mathbb{Q}\), it turns into the torsion radical). Given any radical \(R\) in the category of Abelian groups, its norm \(\|R\|\) is defined to be the least cardinal \(\kappa\) for which \(R\) does not always commute with \(\kappa\)-direct products of groups. If this cardinal exists it has to be regular. Assuming GCH, if \(\kappa\) is any uncountable regular cardinal with the property that no cardinal \(<\kappa\) is weakly compact, then there exists a torsion-free group \(G\) of cardinality \(2^\kappa\), with \(\|R\|=\kappa\). The coslender radical is a non-trivial example of a radical with norm equal to the first non-measurable cardinal [see \textit{R. Dimitrić}, Proc. Conf., Hobart/Aust. 1987, Pitman Res. Notes Math. Ser. 204, 41-50 (1989; Zbl 0671.20053)].
Mathematics Subject Classification ID: 20K27
Mathematics Subject Classification ID: 20K25
Mathematics Subject Classification ID: 20K30
Mathematics Subject Classification ID: 03E50
zbMATH DE Number: 1309628
zbMATH Keywords: categories of Abelian groups
zbMATH Keywords: group radicals
zbMATH Keywords: radicals commuting with infinite products
zbMATH Keywords: rigid systems
zbMATH Keywords: reduced products
zbMATH Keywords: radical norms
zbMATH Keywords: regular cardinals
reviewed by: Radoslav M. Dimitrić
MaRDI profile type: Publication
full work available at URL: s000130050275
OpenAlex ID: W2038254775
DOI: 10.1007/S000130050275
description: scientific article; zbMATH DE number 1309628