Advertisement

Ukrainian Mathematical Journal

, Volume 63, Issue 5, pp 690–698 | Cite as

On generalized derivations satisfying certain identities

  • E. Albaş
Article
Let R be a prime ring with char R ≠ 2 and let d be a generalized derivation on R. We study the generalized derivation d satisfying any of the following identities:
  1. (i)

    d[(x, y)] = [d(x), d(y)] for all x , yR ;

     
  2. (ii)

    d[(x, y)] = [d(y), d(x)] for all x , yR ;

     
  3. (iii)

    either d([x, y]) = [d(x), d(y)] or d([x, y]) = [d(y), d(x)] for all x , yR .

     

Keywords

Prime Ring Generalize Derivation Proper Subgroup Quotient Ring Semiprime Ring 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    E. Albaş, N. Argaç, and V. De Filipps, “Generalized derivations with Engel conditions on one-sided ideals,” Commun. Algebra, 36, No. 6, 2063–2071 (2008).MATHCrossRefGoogle Scholar
  2. 2.
    E. Albaş and N. Argaç, “Generalized derivations of prime rings,” Algebra Colloq., 11, No. 3, 399–410 (2004).MathSciNetMATHGoogle Scholar
  3. 3.
    M. Ashraf, A. Ali, and S. Ali, “Some commutativity theorems for rings with generalized derivations,” Southeast Asian Bull. Math., 31, 415–421 (2007).MathSciNetMATHGoogle Scholar
  4. 4.
    M. Ashraf, A. Ali, and R. Rani, “On generalized derivations of prime rings,” Southeast Asian Bull. Math., 29, No. 4, 669–675 (2005).MathSciNetMATHGoogle Scholar
  5. 5.
    K. I. Beidar, “Rings of quotients of semiprime rings,” Vestn. Mosk. Univ., Ser. I., Mekh., 33, 36–42 (1978), English translation: Transl. Moscow Univ. Math. Bull., 33, 29–34 (1978).MathSciNetMATHGoogle Scholar
  6. 6.
    K. I. Beidar, W. S. Martindale, and V. Mikhalev, “Rings with generalized identities,” Pure Appl. Math. (1996).Google Scholar
  7. 7.
    H. E. Bell and N. Rehman, “Generalized derivations with commutativity and anticommutativity conditions,” Math. J. Okayama Univ., 49, 139–147 (2007).MathSciNetMATHGoogle Scholar
  8. 8.
    M. Brešar, “On the distance of the composition of two derivations to the generalized derivations,” Glasgow Math. J., 33, 89–93 (1991).MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    C. L. Chuang, “GPIs having coefficients in Utumi quotient rings,” Proc. Amer. Math. Soc., 103, No. 3, 723–728 (1988).MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    B. Hvala, “Generalized derivations in prime rings,” Commun. Algebra, 26, No. 4, 1147–1166 (1998).MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    V. K. Kharchenko, “Differential identities of prime rings,” Algebra Logic, 17, 155–168 (1978).MATHCrossRefGoogle Scholar
  12. 12.
    T. K. Lee, “Semiprime rings with differential identities,” Bull. Inst. Math. Acad. Sinica, 20, No. 1, 27–38 (1992).MathSciNetMATHGoogle Scholar
  13. 13.
    T. K. Lee, “Generalized derivations of left faithful rings,” Commun. Algebra, 27, No. 8, 4057–4073 (1999).MATHCrossRefGoogle Scholar
  14. 14.
    T. K. Lee and W. K. Shiue, “Identities with generalized derivations,” Commun. Algebra, 29, No. 10, 4435–4450 (2001).MathSciNetCrossRefGoogle Scholar
  15. 15.
    W. S. Martindale, “Prime rings satisfying a generalized polynomial identity,” J. Algebra, 12, 579–584 (1969).MathSciNetCrossRefGoogle Scholar
  16. 16.
    E. C. Posner, “Derivations in prime rings,” Proc. Amer. Math. Soc., 8, 1093–1100 (1957).MathSciNetCrossRefGoogle Scholar
  17. 17.
    H. Shuliang, “Generalized derivations of prime rings,” Int. J. Math. Math. Sci., 2007, 1–6 (2007).CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2011

Authors and Affiliations

  • E. Albaş
    • 1
  1. 1.IzmirTurkey

Personalised recommendations