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DECOMPOSING LINEAR TRANSFORMATIONS

Part of: Rings and algebras arising under various constructions

Published online by Cambridge University Press:  14 September 2010

LU WANG  and
YIQIANG ZHOU
LU WANG
Affiliation:
Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, Nfld A1C 5S7, Canada (email: lu.wang@mun.ca)
YIQIANG ZHOU*
Affiliation:
Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, Nfld A1C 5S7, Canada (email: zhou@mun.ca)
*
✉For correspondence; e-mail: zhou@mun.ca
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Abstract

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Let R be the ring of linear transformations of a right vector space over a division ring D. Three results are proved: (1) if |D|>4, then for any a∈R there exists a unit u of R such that a+u,a−u and a−u−1 are units of R; (2) if |D|>3 , then for any a∈R there exists a unit u of R such that both a+u and a−u−1 are units of R; (3) if |D|>2 , then for any a∈R there exists a unit u of R such that both a−u and a−u−1 are units of R. The second result extends the main result in H. Chen, [‘Decompositions of countable linear transformations’, Glasg. Math. J. (2010), doi:10.1017/S0017089510000121] and the third gives an affirmative answer to the question raised in the same paper.

Keywords

linear transformationunitunit 1-stable range

MSC classification

Secondary: 16S50: Endomorphism rings; matrix rings
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 83 , Issue 2 , April 2011 , pp. 256 - 261
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2010

Footnotes

The work was supported by a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada.

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