On additive commutator groups in division rings
type: Item
instance of: scholarly article
title: On additive commutator groups in division rings
zbMATH Open document ID: 0894.16010
DOI: 10.1007/BF03322065
author: M. Ariannejad
author: M. L. Mehrabadi
author: S. Akbari
published in: Results in Mathematics
publication date: 1998-09-07T00:00:00Z
review text: Let \(D\) be a division ring with centre \(F\). The authors study algebraicity of \(D/F\) in terms of \([D,D]\), the additive group generated by the additive commutators. If \(D\) is finite-dimensional over \(F\), the reduced trace shows that \(D/[D,D]\) is an \(F\)-space of dimension at most one. In characteristic zero this is also shown to hold when \(D\) is algebraic over \(F\). Next, the authors prove the following generalization of Jacobson's theorem [cf. \textit{N. Jacobson}, Ann. Math., II. Ser. 46, 695-707 (1945; Zbl 0060.07501)]: If for any \(x,y\in D\), \(xy-yx\) is algebraic of degree at most \(n\) (where \(n\) is fixed), then \([D:F]\leq(n+1)^4/4\). They conjecture that here the last bound may be replaced by \(n^2\). The rest of the paper brings a number of special results on separable subfields and primitive elements. Sample: If \(K\) is a finite separable extension of \(F\) in \(D\) without a primitive element in \([D,D]\), then \(K\) has exactly one maximal subfield lying in \([D,D]\).
Mathematics Subject Classification ID: 16K40
Mathematics Subject Classification ID: 16U99
Mathematics Subject Classification ID: 12E15
zbMATH DE Number: 1140101
zbMATH Keywords: division rings
zbMATH Keywords: centre
zbMATH Keywords: algebraicity
zbMATH Keywords: additive commutators
zbMATH Keywords: reduced traces
zbMATH Keywords: Jacobson's theorem
zbMATH Keywords: separable subfields
zbMATH Keywords: primitive elements
zbMATH Keywords: maximal subfields
reviewed by: Paul M. Cohn
MaRDI profile type: Publication
cites work: Elements of reduced trace \(0\)
cites work: Q4126487
cites work: Q3963168
cites work: Commutators in Division Rings
cites work: Algebraic valued functions on noncommutative rings
cites work: Structure theory for algebraic algebras of bounded degree
cites work: A Theorem on Division Rings
cites work: Q3999650
cites work: Extending valuations to algebraic division algebras
cites work: On derived groups of division rings II
full work available at URL: https://doi.org/10.1007/bf03322065
OpenAlex ID: W2061386559
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description: scientific article; zbMATH DE number 1140101