Ukrainian Mathematical Journal

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

On generalized derivations satisfying certain identities

  • E. Albaş
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 .



Prime Ring Generalize Derivation Proper Subgroup Quotient Ring Semiprime Ring 
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  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).zbMATHCrossRefGoogle Scholar
  2. 2.
    E. Albaş and N. Argaç, “Generalized derivations of prime rings,” Algebra Colloq., 11, No. 3, 399–410 (2004).MathSciNetzbMATHGoogle 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).MathSciNetzbMATHGoogle 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).MathSciNetzbMATHGoogle 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).MathSciNetzbMATHGoogle 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).MathSciNetzbMATHGoogle 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).MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    C. L. Chuang, “GPIs having coefficients in Utumi quotient rings,” Proc. Amer. Math. Soc., 103, No. 3, 723–728 (1988).MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    B. Hvala, “Generalized derivations in prime rings,” Commun. Algebra, 26, No. 4, 1147–1166 (1998).MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    V. K. Kharchenko, “Differential identities of prime rings,” Algebra Logic, 17, 155–168 (1978).zbMATHCrossRefGoogle Scholar
  12. 12.
    T. K. Lee, “Semiprime rings with differential identities,” Bull. Inst. Math. Acad. Sinica, 20, No. 1, 27–38 (1992).MathSciNetzbMATHGoogle Scholar
  13. 13.
    T. K. Lee, “Generalized derivations of left faithful rings,” Commun. Algebra, 27, No. 8, 4057–4073 (1999).zbMATHCrossRefGoogle 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

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© Springer Science+Business Media, Inc. 2011

Authors and Affiliations

  • E. Albaş
    • 1
  1. 1.IzmirTurkey

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