Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-10T16:37:23.955Z Has data issue: false hasContentIssue false
  • English
  • Français

Factorization of invertible matrices over rings of stable rank one

Part of: Basic linear algebra Other groups of matrices

Published online by Cambridge University Press:  09 April 2009

Leonid N. Vaserstein  and
Ethel Wheland
Leonid N. Vaserstein
Affiliation:
The Pennsylvania State UniversityUniversity Park, Pennsylvania 16802, U.S.A.
Ethel Wheland
Affiliation:
The Pennsylvania State UniversityUniversity Park, Pennsylvania 16802, U.S.A.
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.

Every invertible n-by-n matrix over a ring R satisfying the first Bass stable range condition is the product of n simple automorphisms, and there are invertible matrices which cannot be written as the products of a smaller number of simple automorphisms. This generalizes results of Ellers on division rings and local rings.

Keywords

stable rankmodulessimple matrices.

MSC classification

Secondary: 15A23: Factorization of matrices 20H25: Other matrix groups over rings
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 48 , Issue 3 , June 1990 , pp. 455 - 460
Copyright
Copyright © Australian Mathematical Society 1990

References

[1]Artin, E., Geometric algebra, (Wiley-Interscience, New York, 1957).Google Scholar
[2]Bourbaki, N., Éléments de mathématique, Livre 2 (Algèbre, Hermann, Paris, 1947).Google Scholar
[3]Dennis, R. K. and Vaserstein, L. N., ‘On a question of M. Newman on the number of commutators’, J. Algebra 118 (1988), 150–161.CrossRefGoogle Scholar
[4]Djoković, D. Ž. and Malzan, J., ‘Products of reflections in the general linear group over a division ring’, Linear Algebra Appl. 28 (1979), 53–62.CrossRefGoogle Scholar
[5]Ellers, E. W., ‘Product of axial affinities and products of central collineations’, in The Geometric Vein, pp. 465–470, (Springer, New York, 1982).Google Scholar
[6]Ellers, E. W. and Ishibashi, H., ‘Factorization of transformations over a local ring’, Linear Algebra 85 (1987), 17–27.CrossRefGoogle Scholar
[7]Ellers, E. W. and Lausch, H., ‘Length theorems for the general linear group of a module over a local ring’, J. Austral. Math. Soc. Ser. A 46 (1989), 122–131.CrossRefGoogle Scholar
[8]Goodearl, K. R. and Menal, P., ‘Stable range one for rings with many units’, J. Pure Appl. Algebra 54 (1988), 261–287.CrossRefGoogle Scholar
[9]Putnam, I. F., ‘The invertible elements are dense in the irrational rotation C*-algebras’, preprint.Google Scholar
[10]Vaserstein, L. N., ‘K1-theory and the congruence subgroup problem’, Mat. Zametki 5 (1969), 233–244 (Translation, Math. Notes 5, 141–148).Google Scholar
[11]Vaserstein, L. N., ‘The stable range of rings and the dimension of topological spaces’, Funkcional. Anal. i Priložen. 5 (1971), 17–27 (Translation, Functional Anal. Appl. 5, 102–110).Google Scholar
[12]Vaserstein, L. N., ‘Bass's first stable range condition’, J. Pure Appl. Algebra 34 (1984), 319–330.CrossRefGoogle Scholar