Decompositions of linear transformations over division rings
type: Item
instance of: scholarly article
zbMATH Open document ID: 1252.15002
DOI: 10.1142/S1005386712000314
author: Huanyin Chen
published in: Algebra Colloquium
publication date: 2012-08-29T00:00:00Z
Mathematics Subject Classification ID: 15A04
Mathematics Subject Classification ID: 16E50
Mathematics Subject Classification ID: 16K40
zbMATH DE Number: 6073846
zbMATH Keywords: linear transformation
zbMATH Keywords: countable dimensional vector space
zbMATH Keywords: automorphism
zbMATH Keywords: division ring
MaRDI profile type: Publication
cites work: COMMUTATIVE RINGS WHOSE ELEMENTS ARE A SUM OF A UNIT AND IDEMPOTENT
cites work: A CHARACTERIZATION OF UNIT REGULAR RINGS
cites work: Exchange rings, units and idempotents
cites work: DECOMPOSITIONS OF COUNTABLE LINEAR TRANSFORMATIONS
cites work: EXTENSIONS OF CLEAN RINGS
cites work: Countable linear transformations are clean
full work available at URL: s1005386712000314
OpenAlex ID: W2147843114
title: Decompositions of linear transformations over division rings
review text: The author proves that for any endomorphism \(\gamma\) of a countable dimensional vector space \(V\) over a division ring \(D\) with more than \(3\) elements, there exist automorphisms \(\alpha\) and \(\beta\) of \(V\) such that \(\gamma-\alpha\), \(\gamma-\alpha^{-1}\), and \(\gamma^2-\beta^2\) are automorphisms of \(V\). In particular, this expresses \(\gamma\) as a sum of two automorphisms in two ways simultaneously. The result generalizes most of the countable dimensional case of a well known theorem of \textit{D. Zelinsky} [Proc. Am. Math. Soc. 5, 627--630 (1954; Zbl 0056.11002)], who proved that, aside from the case of the identity map on a 2-element vector space, every endomorphism of a vector space over a division ring is a sum of two automorphisms. The author also proves that for \(V\), \(D\), and \(\gamma\) as above, there exists an automorphism \(\alpha\) of \(V\) such that \(\gamma+\alpha\) and \(\gamma-\alpha^{-1}\) are automorphisms (cf. [\textit{H. Chen}, Glasg. Math. J. 52, No. 3, 427--433 (2010; Zbl 1218.15002)]).
reviewed by: Kenneth R. Goodearl
description: scientific article; zbMATH DE number 6073846