Free subgroups and free subsemigroups of division rings
type: Item
instance of: scholarly article
title: Free subgroups and free subsemigroups of division rings
zbMATH Open document ID: 0865.16014
author: Katsuo Chiba
published in: Journal of Algebra
publication date: 1997-07-07T00:00:00Z
review text: There are several open questions about free sub-objects of a multiplicative group of a (non-commutative) skew field. Namely, does it contain a non-commutative free subsemigroup (L. Makar-Limanov, 1984) or even a non-commutative free group (A. Lichtman, 1977)? More recently (1992) A. Klein asked whether a multiplicative semigroup of a non-commutative domain contains a non-commutative free subsemigroup. The author answers these questions under some restrictions and improves previously known results. As a sample let me quote one of his results: If \(D\) is a non-commutative division ring with uncountable center \(K\) then \(D^*\) contains a non-commutative free subgroup of the same cardinality as \(K\). The reviewer would also like to mention a similar additive conjecture which contains a question on subsemigroups and has not attracted enough attention: Conjecture. If \(D\) is a finitely generated non-commutative division ring which is infinite dimensional over its center then it contains a non-commutative free subalgebra. (Also submitted to MR).
Mathematics Subject Classification ID: 16K40
Mathematics Subject Classification ID: 20M25
Mathematics Subject Classification ID: 12E15
Mathematics Subject Classification ID: 16U60
zbMATH DE Number: 930157
zbMATH Keywords: multiplicative groups of skew fields
zbMATH Keywords: free subsemigroups
zbMATH Keywords: division rings
zbMATH Keywords: finitely generated non-commutative division rings
zbMATH Keywords: free subalgebras
reviewed by: Leonid Makar-Limanov
MaRDI profile type: Publication
full work available at URL: jabr.1996.0275
OpenAlex ID: W1991284818
DOI: 10.1006/JABR.1996.0275
description: scientific article; zbMATH DE number 930157