Malcev-Neumann series and the free field
type: Item
instance of: scholarly article
title: Malcev-Neumann series and the free field
zbMATH Open document ID: 0951.16007
author: Christophe Reutenauer
published in: Expositiones Mathematicae
publication date: 2000-12-07T00:00:00Z
review text: Let \(R=k\langle X\rangle\) be the free associative algebra in \(X\) over a field \(k\) and \(F\) the skew field of all Malcev-Neumann series in the free group \(\Gamma\) on \(X\) over \(k\). There is a natural embedding of \(R\) in \(F\) and \textit{J. Lewin} has shown [Trans. Am. Math. Soc. 192, 339-346 (1974; Zbl 0307.16009)] that the subfield of \(F\) generated by the image of \(R\) is just the universal field of fractions of \(R\). Lewin's proof used a result of \textit{I. Hughes} [Commun. Pure Appl. Math. 23, 181-188 (1970; Zbl 0214.05401)]. Here the author gives an independent proof, by associating a graph with \(\Gamma\) and using it to analyse a row vector over \(F\) which annihilates a matrix over \(R\).
Mathematics Subject Classification ID: 16S10
Mathematics Subject Classification ID: 16E60
Mathematics Subject Classification ID: 20E05
Mathematics Subject Classification ID: 16U20
zbMATH DE Number: 1417999
zbMATH Keywords: free associative algebras
zbMATH Keywords: Malcev-Neumann series
zbMATH Keywords: free groups
zbMATH Keywords: universal fields of fractions
zbMATH Keywords: graphs
reviewed by: Paul M. Cohn
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1417999