Skip to main content
SpringerLink
Log in

The unitary closure property of the prime varieties of associative algebras

  • Published:
Siberian Mathematical Journal Aims and scope Submit manuscript

Abstract

We prove that every prime variety of associative algebras over an infinite field of characteristic p>0 is generated by either a unital algebra or a nilalgebra of bounded index. We show that the Engel verbally prime T-ideals remain verbally prime as we impose the identity \( x^{p^N } = 0 \) for sufficiently large N. We then describe all prime varieties in an interesting class of varieties of associative algebras.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Kemer A., “On the multilinear components of the regular prime varieties” in: Methods in Ring Theory: Proceedings of the Trento Conference, Marcel Dekker, New York, 1998, pp. 171–183 (Lecture Notes Pure Appl. Math.; 198).

    Google Scholar 

  2. Belov A. Ya., “No associative PI-algebra coincides with its commutant” Siberian Math. J., 44, No. 6, 969-980 (2003).

  3. Samoĭlov L. M., “Algebraic algebras and prime varieties of associative algebras” Sb.: Math., 200, No. 5, 723–751 (2009).

    Article  MATH  MathSciNet  Google Scholar 

  4. Razmyslov Yu. P., “Trace identities of full matrix algebras over a field of characteristic zero” Math. USSR-Izv., 8, No. 4, 727–760 (1974).

    Article  MATH  Google Scholar 

  5. Kemer A., “Multilinear components of the prime subvarieties of the variety Var(M2(F))” Algebr. Represent. Theory, 4, No. 1, 87–104 (2001).

    Article  MATH  MathSciNet  Google Scholar 

  6. Kemer A., “Remarks on the prime varieties” Israel J. Math., 96, No. 2, 341–356 (1996).

    Article  MATH  MathSciNet  Google Scholar 

  7. Samoĭlov L. M., “On the γ-classical varieties” Fundam. Prikl. Mat., 8, No. 3, 887–910 (2002).

    MATH  MathSciNet  Google Scholar 

  8. Kemer A. R., Ideal of Identities of Associative Algebras, Amer. Math. Soc., Providence (1991) (Transl. Math. Monogr.; V. 7).

    Google Scholar 

  9. Kemer A. R., “Identities of finitely generated algebras over an infinite field” Math. USSR, Izv., 37, No. 1, 69–96 (1991).

    Article  MathSciNet  Google Scholar 

  10. Samoĭlov L. M., “Analog of the Levitzki theorem for infinitely generated associative algebras” Math. Notes, 86, No. 1, 143–145 (2009).

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Ulyanovsk State University, Ulyanovsk, Russia

    L. M. Samoĭlov

Authors
  1. L. M. Samoĭlov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to L. M. Samoĭlov.

Additional information

Original Russian Text Copyright © 2010 Samoĭlov L. M.

The author was supported by the Russian Foundation for Basic Research (Grant 07-01-00080).

Ulyanovsk. Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 51, No. 4, pp. 890–903, July–August, 2010.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Samoĭlov, L.M. The unitary closure property of the prime varieties of associative algebras. Sib Math J 51, 712–722 (2010). https://doi.org/10.1007/s11202-010-0072-x

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11202-010-0072-x

Keywords

Search

Navigation