On division rings with algebraic commutators of bounded degree.
type: Item
instance of: scholarly article
title: On division rings with algebraic commutators of bounded degree.
zbMATH Open document ID: 1054.16012
author: Y. Fong
author: P.-H. Lee
author: M. A. Chebotar
published in: Manuscripta Mathematica
publication date: 2004-07-01T00:00:00Z
full work available at URL: 08.pdf
review text: Using results on generalized polynomial and rational identities, the authors prove results about algebras with algebraic commutators. One result is that if \(R\) is a prime ring with extended centroid \(C\), and if for a positive integer \(n\) and all \(x,y\in R\), \(xy-yx\) is algebraic over \(C\) of degree at most \(n\), then \([RC:C]\leq n^2\). For the other results, let \(D\) be a division ring of characteristic \(0\) with center \(Z\) and \(n\) a positive integer. Then \([D:Z]\leq n^2\) if the following elements are algebraic over \(Z\) of degree at most \(n\): all \(E(x)\) for \(x\in D\) and \(E\) a nonzero derivation of \(D\); all \(axa^{-1}x^{-1}\) for \(x\in D\) and \(a\) noncentral in \(D\); or all \(xyx^{-1}y^{-1}\) for \(x,y\in D\) (no characteristic assumption here).
reviewed by: Charles Lanski
Mathematics Subject Classification ID: 16K20
Mathematics Subject Classification ID: 16P10
Mathematics Subject Classification ID: 16U70
Mathematics Subject Classification ID: 16R50
Mathematics Subject Classification ID: 16N60
zbMATH DE Number: 2078006
zbMATH Keywords: algebraic commutators
zbMATH Keywords: polynomial identities
zbMATH Keywords: prime rings
zbMATH Keywords: extended centroids
zbMATH Keywords: derivations
MaRDI profile type: Publication
OpenAlex ID: W2042510044
DOI: 10.1007/S00229-003-0430-0
description: scientific article; zbMATH DE number 2078006