Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-26T23:00:37.794Z Has data issue: false hasContentIssue false
  • English
  • Français

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: [email protected])
YIQIANG ZHOU*
Affiliation:
Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, Nfld A1C 5S7, Canada (email: [email protected])
*
✉For correspondence; e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

References

[1]Camillo, V. P., Khurana, D., Lam, T. Y., Nicholson, W. K. and Zhou, Y., ‘Continuous modules are clean’, J. Algebra 304(1) (2006), 94–111.CrossRefGoogle Scholar
[2]Camillo, V. P. and Yu, H. P., ‘Exchange rings, units and idempotents’, Comm. Algebra 22(12) (1994), 4737–4749.CrossRefGoogle Scholar
[3]Chen, H., ‘Units, idempotents, and stable range conditions’, Comm. Algebra 29(2) (2001), 703–717.CrossRefGoogle Scholar
[4]Chen, H., ‘Decompositions of countable linear transformations’, Glasg. Math. J. (2010), doi:10.1017/S0017089510000121.CrossRefGoogle Scholar
[5]Goodearl, K. R. and Menal, P., ‘Stable range one for rings with many units’, J. Pure Appl. Algebra 54 (1988), 261–287.CrossRefGoogle Scholar
[6]Menal, P. and Moncasi, J., ‘K 1 of von Neumann regular rings’, J. Pure Appl. Algebra 33 (1984), 295–312.CrossRefGoogle Scholar
[7]Utumi, Y., ‘On continuous rings and self injective rings’, Trans. Amer. Math. Soc. 118 (1965), 158–173.CrossRefGoogle Scholar
[8]Wu, T. S., ‘Unit 1-stable range condition’, Chinese Ann. Math. A 16(6) (1995), 760–768.Google Scholar
[9]Zelinsky, D., ‘Every linear transformation is a sum of nonsingular ones’, Proc. Amer. Math. Soc. 5 (1954), 627–630.CrossRefGoogle Scholar
[10]Zhou, Y., ‘On clean group rings’, in: Advances in Ring Theory, Trends in Mathematics (Birkhäuser, Basel, 2009), pp. 335–345.Google Scholar