Advertisement

The Kostrikin Radical and Similar Radicals of Lie Algebras

  • A. Yu. GolubkovEmail author
Article

Abstract

The existing notion of the Kostrikin radical as a radical in the Kurosh–Amitsur sense on classes of Mal’tsev algebras over rings with 1/6 is not completely justified. More precisely, to the fullest extent it is true for classes of Lie algebras over fields of characteristic zero and, as shown in the given paper, classes of algebraic Lie algebras of degree not greater than n over rings with 1/n! at all n ≥ 1. Similar conclusions are obtained in the paper also for the Jordan, regular, and extremal radicals constructed analogously to the Kostrikin radical.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    R. K. Amayo and I. N. Stewart, Infinite Dimensional Lie Algebras, Noordhoff, Leyden (1974).Google Scholar
  2. 2.
    V. A. Andrunakievich and V. M. Ryabukhin, Radicals of Algebras and Structure Theory [in Russian],Nauka, Moscow (1979).zbMATHGoogle Scholar
  3. 3.
    K. I. Beidar and S. A. Pikhtil’kov, “On prime radical of special Lie algebras,” Usp. Mat. Nauk, 49, No. 1, 233 (1994).MathSciNetzbMATHGoogle Scholar
  4. 4.
    G. Benkart, “On inner ideals and ad-nilpotent elements of Lie algebras,” Trans. Am. Math. Soc., 232, 61–81 (1977).MathSciNetCrossRefGoogle Scholar
  5. 5.
    C. Draper Fontanals, A. Fernández López, E. García, and M. Gómez Lozano, “The socle of a non-degenerate Lie algebra,” J. Algebra, 319, 2372–2394 (2008).MathSciNetCrossRefGoogle Scholar
  6. 6.
    A. Fernández López, E. García, and M. Gómez Lozano, “The Jordan algebra of a Lie algebra,” J. Algebra, 308, 164–177 (2007).MathSciNetCrossRefGoogle Scholar
  7. 7.
    A. Fernández López, E. García, and M. Gómez Lozano, “Inner ideal structure of nearly Artinian Lie algebras,” Proc. Am. Math. Soc., 137, No. 1, 1–9 (2009).MathSciNetCrossRefGoogle Scholar
  8. 8.
    A. Fernández López and A. Yu. Golubkov, “Lie algebras with an algebraic adjoint representation revisited,” Manuscripta Math., 140, No. 3-4, 363–376 (2013).MathSciNetCrossRefGoogle Scholar
  9. 9.
    V. T. Filippov, “On nil-elements of index 2 in Mal’tsev algebras,” Sib. Mat. Zh., 22, No. 3 (1981).Google Scholar
  10. 10.
    E. García and M. Gómez Lozano, “A characterization of the Kostrikin radical of a Lie algebra,” J. Algebra, 346, 266–283 (2011).MathSciNetCrossRefGoogle Scholar
  11. 11.
    E. García and M. Gómez Lozano, “A note on a result of Kostrikin,” J. Algebra, 37, 2405–2409 (2009).MathSciNetzbMATHGoogle Scholar
  12. 12.
    A. Yu. Golubkov, “Local finiteness of algebras,” Fundam. Prikl. Mat., 19, No. 6, 25–75 (2014).Google Scholar
  13. 13.
    A. Yu. Golubkov, “Constructions of special radicals of algebras,” Fundam. Prikl. Mat., 20, No. 1, 57–133 (2015).MathSciNetGoogle Scholar
  14. 14.
    A. N. Grishkov, “Local nilpotency of an ideal of a Lie algebra generated by second-order elements,” Sib. Mat. Zh., 23, No. 1, 181–182 (1982).MathSciNetzbMATHGoogle Scholar
  15. 15.
    A. I. Kostrikin, “Lie rings satisfying Engel’s condition,” Izv. Akad. Nauk SSSR. Ser. Mat., 21, 515–540 (1957).MathSciNetzbMATHGoogle Scholar
  16. 16.
    A. I. Kostrikin, “On Burnside’s problem,” Izv. Akad. Nauk SSSR. Ser. Mat., 23, 3–34 (1959).MathSciNetzbMATHGoogle Scholar
  17. 17.
    A. I. Kostrikin, Around Burnside. A Series of Modern Surveys in Mathematics, Springer, Berlin (1990).Google Scholar
  18. 18.
    B. I. Plotkin, “On algebraic sets of elements in groups and Lie algebras,” Usp. Mat. Nauk, 13, No. 6 (84), 133–138 (1958).Google Scholar
  19. 19.
    K. K. Schukin, “RI*-solvable radical of groups,” Mat. Sb., 52, No. 4, 1024–1031 (1966).MathSciNetGoogle Scholar
  20. 20.
    G. B. Seligman, Modular Lie Algebras, Ergebn. Math. ihrer Grenzgeb., Bd. 40, Springer, New York (1967).Google Scholar
  21. 21.
    A. Thedy, “Radicals of right-alternative and Jordan rings,” Commun. Algebra, 12, No. 7, 857–887 (1984).MathSciNetCrossRefGoogle Scholar
  22. 22.
    A. Thedy, “Z-closed ideals of quadratic Jordan algebras,” Commun. Algebra, 13, No. 12, 2537–2565 (1985).MathSciNetCrossRefGoogle Scholar
  23. 23.
    E. I. Zel’manov, “Lie algebras with an algebraic adjoint representation,” Mat. Sb., 121 (163), No. 4 (8), 545–561 (1983).Google Scholar
  24. 24.
    E. I. Zel’manov, “A characterization of the McCrimmon radical,” Sib. Mat. Zh., 25, No. 5, 190–192 (1984).MathSciNetzbMATHGoogle Scholar
  25. 25.
    E. I. Zel’manov and A. I. Kostrikin, “Theorem on sandwich algebras,” Tr. Mat. Inst. Steklova, 183, 106–111 (1988).MathSciNetzbMATHGoogle Scholar
  26. 26.
    K. A. Zhevlakov and I. P. Shestakov, “On local finiteness in the sense of Shirshov,” Algebra Logika, 12, No. 1, 41–73 (1973).MathSciNetCrossRefGoogle Scholar
  27. 27.
    K. A. Zhevlakov, A. M. Slin’ko, I. P. Shestakov, and A. I. Shirshov, Rings That Are Nearly Associative, Academic Press, New York (1982).zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Faculty of Informatics and Control SystemsBauman Moscow State Technical UniversityMoscowRussia

Personalised recommendations