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A result on generalized derivations on right ideals of prime rings

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Ukrainian Mathematical Journal Aims and scope

Let R be a prime ring of characteristic other than 2 and let I be a nonzero right ideal of R. Also let U be the right Utumi quotient ring of R and let C be the center of U. If G is a generalized derivation of R such that [[G(x), x], G(x)] = 0 for all xI, then R is commutative or there exist a, bU such that G(x) = ax + xb for all xR and one of the following assertions is true:

  • (1) (a - λ)I = (0) = (b + λ)I for some λ ∈ C,

  • (2) (a - λ)I = (0) for some λ ∈ C and bC.

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References

  1. E. Albaş, N. Argaç, and V. De Filippis, “Generalized derivations with Engel conditions on one-sided ideals,” Commun. Algebra, 36, No. 6, 2063–2071 (2008).

    Article  MATH  Google Scholar 

  2. K. I. Beidar, W. S. Martindale, III, and A. V. Mikhalev, Rings with Generalized Identities, Marcel Dekker, New York (1996).

    MATH  Google Scholar 

  3. M. Bresar, “One-sided ideals and derivations of prime rings,” Proc. Am. Math. Soc., 122, No. 4, 979–983 (1994).

    MathSciNet  MATH  Google Scholar 

  4. C. L. Chuang, “GPI’s having coefficients in Utumi quotient rings,” Proc. Am. Math. Soc., 103, No. 3, 723–728 (1988).

    Article  MathSciNet  MATH  Google Scholar 

  5. B. Felzenszwalb, “Derivations in prime rings,” Proc. Am. Math. Soc., 84, No. 1, 16–20 (1982).

    Article  MathSciNet  MATH  Google Scholar 

  6. V. De Filippis and M. S. Tammam El-Sayiad, “A note on Posner’s theorem with generalized derivations on Lie ideals,” Rend. Semin. Mat. Univ. Padova, 122, 55–64 (2009).

    MathSciNet  MATH  Google Scholar 

  7. V. De Filippis, “Generalized derivations in prime rings and noncommutative Banach algebras,” Bull. Korean Math. Soc., 45, No. 4, 621–629 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  8. J. S. Erickson, W. S. Martindale, III, and J. M. Osborn, “Prime nonassociative algebras,” Pacif. J. Math., 60, 49–63 (1975).

    MathSciNet  MATH  Google Scholar 

  9. I. N. Herstein, Topics in Ring Theory, Univ. Chicago Press, Chicago (1969).

    MATH  Google Scholar 

  10. B. Hvala, “Generalized derivations in rings,” Commun. Algebra, 26(4), 1147–1166 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  11. V. K. Kharchenko, “Differential identities of prime rings,” Algebra Logic, 17, 155–168 (1978).

    Article  MATH  Google Scholar 

  12. T. K. Lee, “Semiprime rings with differential identities,” Bull. Inst. Math. Acad. Sinica, 20, No. 1, 27–38 (1992).

    MathSciNet  MATH  Google Scholar 

  13. T. K. Lee, “Generalized derivations of left faithful rings,” Commun. Algebra, 27, No. 8, 4057–4073 (1999).

    Article  MATH  Google Scholar 

  14. T. K. Lee and W. K. Shiue, “Identities with generalized derivations,” Commun. Algebra, 29, No. 10, 4435–4450 (2001).

    Article  MathSciNet  Google Scholar 

  15. W. S. Martindale III, “Prime rings satisfying a generalized polynomial identity,” J. Algebra, 12, 576–584 (1969).

    Article  MathSciNet  MATH  Google Scholar 

  16. A. Richoux, “A theorem for prime rings,” Proc. Am. Math. Soc., 77, No. 1, 27–31 (1979).

    Article  MathSciNet  MATH  Google Scholar 

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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 64, No. 2, pp. 165–175, February, 2012.

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Demir, Ç., Argaç, N. A result on generalized derivations on right ideals of prime rings. Ukr Math J 64, 186–197 (2012). https://doi.org/10.1007/s11253-012-0637-x

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  • DOI: https://doi.org/10.1007/s11253-012-0637-x

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