On group rings of nilpotent groups
type: Item
instance of: scholarly article
title: On group rings of nilpotent groups
zbMATH Open document ID: 0556.16006
DOI: 10.1007/BF02761167
published in: Israel Journal of Mathematics
publication date: 1984-01-01T00:00:00Z
review text: Let F be a field and G a torsion-free non-abelian nilpotent group. It is well known that G contains a subgroup \(H=<a,b;\quad [a,b]=c,\quad [a,c]=[b,c]=1>.\) The group algebras F[H] and F[G] are Ore domains and their skew fields of fractions are denoted by F(H) and F(G), respectively. It is proved that the elements \((1-a)^{-1}\) and \((1- a)^{-1}(1-b)^{-1}\) generate a free subalgebra of F(H) of rank 2. It follows that F(H), and hence F(G), contains a free subalgebra of any countable rank.
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 16S10
Mathematics Subject Classification ID: 20F18
Mathematics Subject Classification ID: 16P50
Mathematics Subject Classification ID: 16Kxx
zbMATH DE Number: 3886073
zbMATH Keywords: torsion-free non-abelian nilpotent group
zbMATH Keywords: group algebras
zbMATH Keywords: Ore domains
zbMATH Keywords: skew fields of fractions
zbMATH Keywords: free subalgebra
author: Leonid Makar-Limanov
reviewed by: Patrick F. Smith
MaRDI profile type: Publication
cites work: Q4126487
cites work: Q5527879
cites work: Groups of polynomial growth and expanding maps. Appendix by Jacques Tits
cites work: The Group Ring Of a Class Of Infinite Nilpotent Groups
cites work: The skew field of fractions of the weyl algebra contains a free noncommutative subalgebra
cites work: Quotient Rings of Group Rings
description: scientific article; zbMATH DE number 3886073