On universal fields of fractions for free algebras
type: Item
instance of: scholarly article
title: On universal fields of fractions for free algebras
zbMATH Open document ID: 0959.17010
author: A. I. Lichtman
published in: Journal of Algebra
publication date: 2001-04-29T00:00:00Z
review text: Let \(K\langle x_i \mid i\in I\rangle_K\) be a free associative algebra over a field \(K\). The universal field of fractions for this algebra was constructed by \textit{S. A. Amitsur} in connection with the study of rational identities in skew fields [J. Algebra 3, 304-359 (1966; Zbl 0203.04003)]. Such skew fields are called also ''free fields''. Now let \(L\) be an arbitrary Lie algebra over a field \(K\) and \(U(L)\) be its universal enveloping algebra. \textit{P. M. Cohn} proved that \(U(L)\) can be embedded in a skew field \(\widetilde{D(L)}\) [Proc. Lond. Math. Soc. (3) 11, 511-530 (1961; Zbl 0104.03203)]. The skew subfield of \(\widetilde{D(L)}\) generated by \(U(L)\) denote by \(D(L)\); in fact \(D(L)\) is a proper skew subfield of \(\widetilde{D(L)}\) and \(\widetilde{D(L)}\) is a topological completion of \(D(L)\). The author suggested another method for the construction of \(D(L)\) [\textit{A. Lichtman}, J. Algebra 177, 870-898 (1995; Zbl 0837.16019)]. Now the main result is as follows. Theorem. Let \(H\) be a free Lie algebra. Then the skew field \(D(H)\) is isomorphic to the universal field of fractions for the free associative algebra \(U(H)\).
Mathematics Subject Classification ID: 17B35
Mathematics Subject Classification ID: 16S10
Mathematics Subject Classification ID: 16K40
Mathematics Subject Classification ID: 16S30
Mathematics Subject Classification ID: 16W60
zbMATH DE Number: 1526430
zbMATH Keywords: skew field
zbMATH Keywords: free field
zbMATH Keywords: Ore domain
zbMATH Keywords: valuation
reviewed by: Victor Petrogradsky
MaRDI profile type: Publication
full work available at URL: jabr.2000.8344
OpenAlex ID: W1970957632
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DOI: 10.1006/JABR.2000.8344
description: scientific article; zbMATH DE number 1526430