Selected Works of A.I. Shirshov pp 55-68 | Cite as

# On some Nonassociative Nil-rings and Algebraic Algebras

Chapter

## 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]A.A. Albert,
*Power associative rings*, Trans. Amer. Math Soc. 64 (1948) 552–593.MATHCrossRefMathSciNetGoogle Scholar - [2]G. Birkhoff,
*Representability of Lie algebras and Lie groups by matrices*, Annals of Math. 38 (1937) 526–632.CrossRefMathSciNetGoogle Scholar - [3]N. Jacobson,
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*Treue Darstellung Liescher Ringe*, J. reine und angew. Math. 177 (1937) 152–160.Google Scholar

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