Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Filtered algebraic algebras

Author: Alon Regev
Journal: Proc. Amer. Math. Soc. 138 (2010), 1941-1947
MSC (2010): Primary 16S15, 16U99
Published electronically: February 9, 2010
MathSciNet review: 2596027
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Small and Zelmanov posed the question whether every element of a graded algebra over an uncountable field must be nilpotent, provided that the homogeneous elements are nilpotent. This question has recently been answered in the negative by Smoktunowicz. In this paper we prove that the answer is affirmative for associated graded algebras of filtered algebraic algebras. Our result is based on Amitsur's theorems on algebras over infinite fields.

References [Enhancements On Off] (What's this?)

  • 1. S.A. Amitsur.
    Algebras over infinite fields.
    Proceedings of the American Mathematical Society, 7(1):35-48, 1956. MR 0075933 (17:822b)
  • 2. S.A. Amitsur.
    Countably generated division algebras over nondenumerable fields.
    Bulletin of the Research Council of Israel, Section F, 7(1):39, 1957.
  • 3. L. Bartholdi.
    Branch rings, thinned rings, tree enveloping rings.
    Israel J. Math., 154:93-139, 2006. MR 2254535 (2007k:20051)
  • 4. G. Bergman.
    Radicals, tensor products and algebraicity.
    Israel Mathematical Conference Proceedings, 1:150-192, 1989. MR 1029309 (91d:16033)
  • 5. L. Small and E. Zelmanov.
    Private communication, 2006.
  • 6. A. Regev.
    Remarks on a theorem of Amitsur.
    Communications in Algebra (to appear).
  • 7. I. Kaplansky.
    On a problem of Kurosch and Jacobson.
    Bull. Amer. Math. Soc., 52(12):1033-1035, 1946. MR 0019600 (8:435a)
  • 8. I. Schur.
    Uber eine Klasse von Matrizen, die sich einergegebenen Matrix zuordenenlassen.
    Dissertation (1901); reprinted in Gesammelte Abhandlungen, Band, I, 1-72, Springer Verlag, Heidelberg, 1973. MR 0462891 (57:2858a)
  • 9. A. Smoktunowicz.
    The Jacobson radical of rings with nilpotent homogeneous elements.
    Bull. London Math. Soc., 40:917-928, 2008. MR 2471940

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 16S15, 16U99

Retrieve articles in all journals with MSC (2010): 16S15, 16U99

Additional Information

Alon Regev
Affiliation: Department of Mathematical Sciences, Watson Hall 320, Northern Illinois University, DeKalb, Illinois 60115

Received by editor(s): April 24, 2009
Received by editor(s) in revised form: August 13, 2009, and September 14, 2009
Published electronically: February 9, 2010
Communicated by: Birge Huisgen-Zimmermann
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society