Skip to main content
SpringerLink
Log in

Algebraic derivations with constants satisfying a polynomial identity

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

We prove that if a semiprime ringR possesses a derivation which is integral over its extended centroidC and whose constants satisfy a polynomial identity, thenR itself is a PI-ring. This answers affirmatively a problem raised by M. Smith in 1975 and recently again by Bergen and Grzeszczuk [4].

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. S. A. Amitsur,A note on PI-rings, Israel Journal of Mathematics10 (1971), 210–211.

    MATH  MathSciNet  Google Scholar 

  2. K. I. Beidar and A. V. Mikhälev,Orthogonal completeness and algebraic systems, Uspekhi Matematicheskikh Nauk40 (1985), 79–115.

    Google Scholar 

  3. K. I. Beidar, W. S. Martindale 3rd and A. V. Mikhälev,Rings with Generalized Identities, Marcel Dekker, Inc., New York-Basel-Hong Kong, 1996.

    MATH  Google Scholar 

  4. J. Bergen and P. Grzeszczuk,Skew derivations whose invariants satisfy a polynomial identity, Journal of Algebra228 (2000), 710–737.

    Article  MATH  MathSciNet  Google Scholar 

  5. C.-L. Chuang,GPIs having coefficients in Utumi quotient rings, Proceedings of the American Mathematical Society103 (1988), 723–728.

    Article  MATH  MathSciNet  Google Scholar 

  6. M. Cohen,Centralizers of algebraic elements, Communications in Algebra6 (1978), 1505–1519.

    Article  MATH  MathSciNet  Google Scholar 

  7. C. Faith and Y. Utumi,On a new proof of Litoff’s theorem, Acta Mathematica Academiae Scientiarum Hungaricae14 (1963), 369–371.

    Article  MATH  MathSciNet  Google Scholar 

  8. I. N. Herstein,Topics in Algebra, 2nd ed., Xerox College Publishing, 1975.

  9. I. N. Herstein,Rings with involution, University of Chicago Press, Chicago, 1976.

    MATH  Google Scholar 

  10. I. N. Herstein and L. Neumann,Centralizers in rings, Annali di Matematica Pura ed Applicata (4)102 (1975), 37–44.

    MathSciNet  Google Scholar 

  11. N. Jacobson,PI-algebras: An Introduction, Lecture Notes in Mathematics441, Springer-Verlag, Berlin and New York, 1975.

    MATH  Google Scholar 

  12. V. K. Kharchenko,Differential identities of prime rings, Algebra i Logika17 (1978), 220–238; English translation: Algebra and Logika17 (1978), 154–168.

    MathSciNet  Google Scholar 

  13. V. K. Kharchenko,Differential identities of semiprime rings, Algebra i Logika18 (1979), 86–119; English translation: Algebra and Logika18 (1979), 58–80.

    MathSciNet  Google Scholar 

  14. V. K. Kharchenko,Automorphisms and Derivations of Associative Rings, Kluwer Academic Publishers, Dordrecht, 1991.

    MATH  Google Scholar 

  15. V. K. Kharchenko, J. Keller and S. Rodrigues-Romo,Prime rings with PI rings of constants, Israel Journal of Mathematics96 (1996), 357–377.

    Article  MATH  MathSciNet  Google Scholar 

  16. T.-K. Lee,Semiprime rings with differential identities, Bulletin of the Institute of Mathematics. Academic Sinica20 (1992), 27–38.

    MATH  Google Scholar 

  17. W. S. Martindale 3rd,Prime rings satisfying a generalized polynomial identity, Journal of Algebra12 (1969), 576–584.

    Article  MATH  MathSciNet  Google Scholar 

  18. S. Montgomery,Centralizers satisfying polynomial identities, Israel Journal of Mathematics18 (1974), 209–219.

    Article  Google Scholar 

  19. L. H. Rowen,Some results on the center of a ring with polynomial identity, Proceedings of the American Mathematical Society79 (1973), 219–223.

    MATH  MathSciNet  Google Scholar 

  20. L. H. Rowen,General polynomial identities. II., Journal of Algebra38 (1976), 380–392.

    Article  MATH  MathSciNet  Google Scholar 

  21. M. K. Smith,Rings with an integral element whose centralizer satisfies a polynomial identity, Duke Mathematical Journal42 (1975), 137–149.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, National Taiwan University, 106, Taipei, Taiwan

    Chen-Lian Chuang & Tsiu-Kwen Lee

Authors
  1. Chen-Lian Chuang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tsiu-Kwen Lee
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Chuang, CL., Lee, TK. Algebraic derivations with constants satisfying a polynomial identity. Isr. J. Math. 138, 43–60 (2003). https://doi.org/10.1007/BF02783418

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02783418

Keywords

Search

Navigation