Mediterranean Journal of Mathematics

, Volume 13, Issue 1, pp 53–64 | Cite as

Power-Commuting Generalized Skew Derivations in Prime Rings

  • Luisa Carini
  • Vincenzo De Filippis
  • Giovanni Scudo
Article

Abstract

Let R be a non-commutative prime ring of characteristic different from 2 with extended centroid C, F ≠ 0 a generalized skew derivation of R, and n ≥ 1 such that [F(x), x] n  = 0, for all xR. Then there exists an element λ ∈ C such that F(x) = λx, for all xR.

Keywords

Prime rings differential identities generalized skew derivations 

Mathematical Subject Classification

16N60 16W25 47B47 47B48 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Beidar, K.I., Martindale, W.S., III, Mikhalev, A.V.: Rings with Generalized Identities, Pure and Applied Mathematics, Dekker, New York (1996)Google Scholar
  2. 2.
    Carini, L., De Filippis, V.: Commutators with power central values on a Lie ideal. Pac. J. Math. 193/2, 296–278 (2000)Google Scholar
  3. 3.
    Chang J.-C.: On the identity h(x) = af(x) + g(x)b. Taiwan. J. Math. 7, 103–113 (2003)MATHGoogle Scholar
  4. 4.
    Chuang C.-L.: GPIs having coefficients in Utumi quotient rings. Proc. Am. Math. Soc. 103, 723–728 (1988)CrossRefMathSciNetMATHGoogle Scholar
  5. 5.
    Chuang C.-L.: Differential identities with automorphisms and antiautomorphisms I. J. Algebra 149, 371–404 (1992)CrossRefMathSciNetMATHGoogle Scholar
  6. 6.
    Chuang C.-L.: Differential identities with automorphisms and antiautomorphisms II. J. Algebra 160, 130–171 (1993)CrossRefMathSciNetMATHGoogle Scholar
  7. 7.
    Chuang C.-L, Chou M.-C, Liu C.-K: Skew derivations with annihilating Engel conditions. J. Publ. Math. Debrecen 68/1–2, 161–170 (2006)MathSciNetGoogle Scholar
  8. 8.
    Chuang C.-L, Lee T.-K: Identities with a single skew derivation. J. Algebra 288, 59–77 (2005)CrossRefMathSciNetMATHGoogle Scholar
  9. 9.
    De Filippis V.: Generalized derivations and commutators with nilpotent values on Lie ideals. Tamsui Oxford J. Math. Sci. 22/2, 167–175 (2006)MathSciNetGoogle Scholar
  10. 10.
    De Filippis V., Di Vincenzo O.M.: Vanishing derivations and centralizers of generalized derivations on multilinear polynomials. Comm. Algebra 40/6, 1918–1932 (2012)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Herstein I.N.: Center-like elements in prime rings. J. Algebra 60/2, 567–574 (1979)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Hvala B.: Generalized derivations in rings. Comm. Algebra 26, 1147–1166 (1998)CrossRefMathSciNetMATHGoogle Scholar
  13. 13.
    Jacobson, N.: Structure of Rings. American Mathematical Society, Providence (1964)Google Scholar
  14. 14.
    Kharchenko V.K.: Generalized identities with automorphisms. Algebra Log. 14, 132–148 (1975)CrossRefGoogle Scholar
  15. 15.
    Lanski C.: Differential identities, Lie ideals and Posner’s theorems. Pac. J. Math. 134/2, 275–297 (1988)CrossRefMathSciNetGoogle Scholar
  16. 16.
    Lee T.-K.: Generalized derivations of left faithful rings. Comm. Algebra 27, 4057–4073 (1999)CrossRefMathSciNetMATHGoogle Scholar
  17. 17.
    Lee T.-K.: Generalized skew derivations characterized by acting on zero products. Pac. J. Algebra 216, 293–301 (2004)MATHGoogle Scholar
  18. 18.
    Lee T.-K, Liu K.-S: Generalized skew derivations with algebraic values of bounded degree. Houston J. Math. 39/3, 733–740 (2013)MathSciNetGoogle Scholar
  19. 19.
    Martindale W.S. III: Prime rings satisfying a generalized polynomial identity. J. Algebra 12, 576–584 (1969)CrossRefMathSciNetMATHGoogle Scholar
  20. 20.
    Posner E.C.: Derivations in prime rings. Proc. Am. Math. Soc. 8, 1093–1100 (1957)CrossRefMathSciNetGoogle Scholar
  21. 21.
    Wang Y.: Power-centralizing automorphisms of Lie ideals in prime rings. Comm. Algebra 34, 609–615 (2006)CrossRefMathSciNetMATHGoogle Scholar

Copyright information

© Springer Basel 2014

Authors and Affiliations

  • Luisa Carini
    • 1
  • Vincenzo De Filippis
    • 1
  • Giovanni Scudo
    • 1
  1. 1.Department of MathematicsUniversity of MessinaMessinaItaly

Personalised recommendations