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Siberian Mathematical Journal

, Volume 58, Issue 3, pp 461–466 | Cite as

Simple finite-dimensional algebras without finite basis of identities

  • A. V. KislitsinEmail author
Article
  • 31 Downloads

Abstract

In 1993, Shestakov posed a problem of existence of a central simple finite-dimensional algebra over a field of characteristic 0 whose identities cannot be defined by a finite set (Dniester Notebook, Problem 3.103). In 2012, Isaev and the author constructed an example that gave a positive answer to this problem. In 2015, the author constructed an example of a central simple seven-dimensional commutative algebra without finite basis of identities. In this article we continue the study of Shestakov’s problem in the case of anticommutative algebras. We construct an example of a simple seven-dimensional anticommutative algebra over a field of characteristic 0 without finite basis of identities.

Keywords

simple algebra identity of algebra basis of identities nonfinitely based algebra strongly nonfinitely based algebra 

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References

  1. 1.
    Vaughan-Lee M. R., “Varieties of Lie algebras,” Q. J. Math., Oxf. II Ser., vol. 21, no. 83, 297–308 (1970).MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Drenski V. S., “On identities in Lie algebras,” Algebra and Logic, vol. 13, no. 3, 150–165 (1974).MathSciNetCrossRefGoogle Scholar
  3. 3.
    Polin S. V., “On identities of finite algebras,” Sib. Math. J., vol. 17, no. 6, 992–999 (1976).MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    L’vov I. V., “Finite-dimensional algebras with infinite bases of identities,” Sib. Math. J., vol. 19, no. 1, 63–69 (1978).CrossRefzbMATHGoogle Scholar
  5. 5.
    Isaev I. M. and Kislitsin A. V., “The identities of vector spaces embedded in a linear algebra,” Sib. Electron. Math. Rep., vol. 12, 328–343 (2015).MathSciNetzbMATHGoogle Scholar
  6. 6.
    Isaev I. M., “Finite-dimensional right alternative algebras that do not generate finitely based varieties,” Algebra and Logic, vol. 25, no. 2, 86–96 (1986).CrossRefzbMATHGoogle Scholar
  7. 7.
    Specht W., “Gesetze in Ringen. I,” Math. Z., vol. 52, no. 1, 557–589 (1950).MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Filippov V. T., Kharchenko V. K., and Shestakov I. P., Dniester Notebook: Unsolved Problems in the Theory of Rings and Modules. 3 ed., Inst. Math., Novosibirsk (1982).zbMATHGoogle Scholar
  9. 9.
    Kemer A. R., “Finite basis property of identities of associative algebras,” Algebra and Logic, vol. 26, no. 5, 362–397 (1987).MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Belov A. Ya., “The local finite basis property and local representability of varieties of associative rings,” Izv. Math., vol. 74, no. 1, 1–126 (2010).MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Isaev I. M., “Essentially infinitely based varieties of algebras,” Sib. Math. J., vol. 30, no. 6, 892–894 (1989).CrossRefzbMATHGoogle Scholar
  12. 12.
    Isaev I. M., “Finite algebras with no independent basis of identities,” Algebra Universalis, vol. 37, no. 4, 440–444 (1997).MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Filippov V. T., Kharchenko V. K., and Shestakov I. P., Dniester Notebook: Unsolved Problems in the Theory of Rings and Modules. 4 ed., Inst. Math., Novosibirsk (1993).zbMATHGoogle Scholar
  14. 14.
    Shestakov I. and Zaicev M., “Polynomial identities of finite dimensional simple algebras,” Comm. Algebra, vol. 39, no. 3, 929–932 (2011).MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Isaev I. M. and Kislitsin A. V., “An example of a simple finite-dimensional algebra with no finite basis of identities,” Dokl. Math., vol. 86, no. 3, 774–775 (2012).MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Isaev I. M. and Kislitsin A. V., “Example of simple finite dimensional algebra with no finite basis of its identities,” Comm. Algebra, vol. 41, no. 12, 4593–4601 (2013).MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Kislitsin A. V., “An example of a central simple commutative finite-dimensional algebra with an infinite basis of identities,” Algebra and Logic, vol. 54, no. 3, 204–210 (2015).MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Parfenov V. A., “Specht’s problem in ε-algebras,” Math. Notes, vol. 34, no. 2, 577–582 (1983).MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Isaev I. M. and Kislitsin A. V., “Identities in vector spaces and examples of finite-dimensional linear algebras having no finite basis of identities,” Algebra and Logic, vol. 52, no. 4, 290–307 (2013).MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2017

Authors and Affiliations

  1. 1.Dostoevsky Omsk State UniversityOmskRussia
  2. 2.Altaĭ State Pedagogical UniversityBarnaulRussia

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