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On some Nonassociative Nil-rings and Algebraic Algebras

  • A. I. Shirshov
Part of the Contemporary Mathematicians book series (CM)

Abstract

In the works of Levitzki [5] and Jacobson [3] devoted to the solution of the problem of Kurosh [4], it is proved that any associative algebraic algebra of bounded degree is locally finite, and that every associative nil-ring of bounded index is locally nilpotent. The problem of Kurosh can be stated for any class of power associative algebras [1], but already Lie algebras give an example showing that the problem does not have a positive solution for arbitrary power associative algebras.

Keywords

Inductive Hypothesis Additive Group Associative Algebra Jordan Algebra Associative Ring 
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References

  1. [1]
    A.A. Albert, Power associative rings, Trans. Amer. Math Soc. 64 (1948) 552–593.MATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    G. Birkhoff, Representability of Lie algebras and Lie groups by matrices, Annals of Math. 38 (1937) 526–632.CrossRefMathSciNetGoogle Scholar
  3. [3]
    N. Jacobson, Structure theory for algebraic algebras of bounded degree, Annals of Math. 46 (1945) 695–707.CrossRefGoogle Scholar
  4. [4]
    A.G. Kurosh, Problems in the theory of rings related to the Burnside problem on periodic groups, Izv. Akad. Nauk USSR Ser. Mat. 5 (1941) 233–240.Google Scholar
  5. [5]
    I. Levitzki, On a problem of A. Kurosch, Bull. Amer. Math. Soc. 52 (1946) 1033–1035.MATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    A.I. Malcev, On algebras defined by identical relations, Mat. Sbornik N.S. 26 (1950) 19–33.MathSciNetGoogle Scholar
  7. [7]
    R.D. Schafer, Representations of alternative algebras, Trans. Amer. Math. Soc. 72 (1952) 1–17.MATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    A.I. Shirshov, On special J-rings, Mat. Sbornik 38 (1956) 149–166.Google Scholar
  9. [9]
    E. Witt, Treue Darstellung Liescher Ringe, J. reine und angew. Math. 177 (1937) 152–160.Google Scholar

Copyright information

© Birkhäuser Verlag, P.O. Box 133, CH-4010 Basel, Switzerland 2009

Authors and Affiliations

  • A. I. Shirshov

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