Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-25T07:41:16.708Z Has data issue: false hasContentIssue false
  • English
  • Français

On transposes of nilpotent matrices over arbitrary rings

Published online by Cambridge University Press:  09 April 2009

Thomas P. Kezlan
Thomas P. Kezlan
Affiliation:
Department of Mathematics, University of Missouri — Kansas City, Kansas City, Missouri 64110, 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.

It is shown that if every nilpotent 2 × 2 matrix over a ring has nilpotent transpose, then the commutator ideal must be contained in the Jacobson radical, thus generalizing a result of R. S. Gupta, who considered the division ring case. Moreover, if the nilpotent elements form an ideal or if the ring satisfies a polynomial identity, then the above property of the transpose implies that in fact the commutator ideal must be nil.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 23 , Issue 3 , May 1977 , pp. 366 - 370
Copyright
Copyright © Australian Mathematical Society 1977

References

Amitsur, S. A. (1957), ‘A generalization of Hilbert's Nullstellensatz’, Proc. Amer. Math. Soc. 8, 649–656, MR 19, 384.Google Scholar
Andrunakievic, V. A. and JuRjabuhin, M. (1968), ‘Rings without nilpotent elements and completely prime ideals’, Soviet Math. Dokl. 9, 565–568, MR 37, 6320.Google Scholar
Awtar, Ram (1973), ‘A remark on the commutativity of certain rings’, Proc. Amer. Math. Soc. 41, 370–372, MR 48, 6184.CrossRefGoogle Scholar
Drazin, M. P. (1955), ‘Engel rings and a result of Herstein and Kaplansky’, Amer. J. Math. 77, 895–913, MR 17, 1048.CrossRefGoogle Scholar
Gupta, R. N. (1970), ‘Nilpotent matrices with invertible transpose’, Proc. Amer. Math. Soc. 24, 572–575, MR 40, 5628.CrossRefGoogle Scholar
Herstein, I. N. (1963), ‘A remark on rings and algebras’, Mich. Math. J. 10, 269–272, MR 28, 2129.CrossRefGoogle Scholar
Jacobson, N., Structure of Rings (rev. ed., Amer. Math. Soc. Colloq. Publ., vol. 37, Amer. Math. Soc., Providence, R.I.).Google Scholar
Kezlan, T. P. (1966), ‘Rings in which certain subsets satisfy polynomial identities’, Trans. Amer. Math. Soc. 125, 414–421, MR 36, 211.CrossRefGoogle Scholar