Burnside's theorem for matrix rings over division rings
type: Item
instance of: scholarly article
title: Burnside's theorem for matrix rings over division rings
zbMATH Open document ID: 1068.15019
author: P. Rosenthal
author: Bamdad R. Yahaghi
author: Mehdi Radjabalipour
published in: Linear Algebra and its Applications
publication date: 2004-08-06T00:00:00Z
review text: The setting for this paper is a left subalgebra \({\mathcal A}\) of the matrix algebra \(M_n(D)\) over a general division ring \(D\) (to wit, \({\mathcal A}\) is a subring of \(M_n(D)\) which is a left \(D\)-module, too). A version of Burnside's theorem is here presented by showing that every transitive (and hence irreducible) left subalgebra of \(M_n(D)\) containing a rank-one matrix is equal to \(M_n(D)\). The last condition compensates for the lack of the requirement that \(D\) be an algebraically closed field. Counterexamples are also given in case \(D\) is the ring \({\mathbf H}\) of real quaternions and \(n\) is any even integer, where \({\mathcal A}\) is irreducible but not transitive, contrary to the commutative case.
reviewed by: George F. Nassopoulos
Mathematics Subject Classification ID: 15A30
Mathematics Subject Classification ID: 15A04
Mathematics Subject Classification ID: 15B33
Mathematics Subject Classification ID: 16S50
zbMATH DE Number: 2081918
zbMATH Keywords: Division ring
zbMATH Keywords: Matrix (left) algebra
zbMATH Keywords: Submodule
zbMATH Keywords: Irreducible (left) algebra
zbMATH Keywords: Burnside's theorem
zbMATH Keywords: Invariant (right) subspace
zbMATH Keywords: counterexamples
zbMATH Keywords: quaternions
MaRDI profile type: Publication
full work available at URL: j.laa.2003.08.011
OpenAlex ID: W2069459603
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DOI: 10.1016/J.LAA.2003.08.011
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description: scientific article; zbMATH DE number 2081918