Skip to main content
SpringerLink
Log in

Generalized derivations on Lie ideals in prime rings

  • Published:
Czechoslovak Mathematical Journal Aims and scope Submit manuscript

Abstract

Let R be a prime ring with its Utumi ring of quotients U and extended centroid C. Suppose that F is a generalized derivation of R and L is a noncentral Lie ideal of R such that F(u)[F(u), u]n = 0 for all u ∈ L, where n ⩾ 1 is a fixed integer. Then one of the following holds:

  1. (1)

    there exists λ ∈ C such that F(x) = λx for all x ∈ R

  2. (2)

    R satisfies s 4 and F(x) = ax + xb for all x ∈ R, with a, b ∈ U and a − b ∈ C

  3. (3)

    char(R) = 2 and R satisfies s 4.

As an application we also obtain some range inclusion results of continuous generalized derivations on Banach algebras.

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. K. I. Beidar, W. S. Martindale III, A. V. Mikhalev: Rings with Generalized Identities. Monographs and Textbooks in Pure and Applied Mathematics 196, Marcel Dekker, New York, 1996.

    MATH  Google Scholar 

  2. J. Bergen, I. N. Herstein, J. W. Kerr: Lie ideals and derivations of prime rings. J. Algebra 71 (1981), 259–267.

    Article  MATH  MathSciNet  Google Scholar 

  3. M. Brešar, J. Vukman: On left derivations and related mappings. Proc. Am. Math. Soc. 110 (1990), 7–16.

    Article  MATH  Google Scholar 

  4. L. Carini, V. De Filippis: Commutators with power central values on a Lie ideal. Pac. J. Math. 193 (2000), 269–278.

    Article  MATH  Google Scholar 

  5. C.-L. Chuang: GPIs having coefficients in Utumi quotient rings. Proc. Am. Math. Soc. 103 (1988), 723–728.

    Article  MATH  MathSciNet  Google Scholar 

  6. V. De Filippis: Generalized derivations and commutators with nilpotent values on Lie ideals. Tamsui Oxf. J. Math. Sci. 22 (2006), 167–175.

    MATH  MathSciNet  Google Scholar 

  7. V. de Filippis, G. Scudo, M. S. Tammam El-Sayiad: An identity with generalized derivations on Lie ideals, right ideals and Banach algebras. Czech. Math. J. 62 (2012), 453–468.

    Article  MATH  Google Scholar 

  8. B. Dhara: Power values of derivations with annihilator conditions on Lie ideals in prime rings. Commun. Algebra 37 (2009), 2159–2167.

    Article  MATH  MathSciNet  Google Scholar 

  9. T. S. Erickson, W. S. Martindale III, J. M. Osborn: Prime nonassociative algebras. Pac. J. Math. 60 (1975), 49–63.

    Article  MATH  MathSciNet  Google Scholar 

  10. B. E. Johnson, A. M. Sinclair: Continuity of derivations and a problem of Kaplansky. Am. J. Math. 90 (1968), 1067–1073.

    Article  MATH  MathSciNet  Google Scholar 

  11. N. Jacobson: Structure of Rings. American Mathematical Society Colloquium Publications 37, American Mathematical Society, Providence, 1964.

    Google Scholar 

  12. V. K. Kharchenko: Differential identities of prime rings. Algebra Logic 17 (1979), 155–168; translation from Algebra i Logika 17 (1978), 220–238, 242–243. (In Russian.)

    Article  MATH  Google Scholar 

  13. B.-D. Kim: Jordan derivations on prime rings and their applications in Banach algebras, I. Commun. Korean Math. Soc. 28 (2013), 535–558.

    Article  MATH  MathSciNet  Google Scholar 

  14. B.-D. Kim: Derivations of semiprime rings and noncommutative Banach algebras. Commun. Korean Math. Soc. 17 (2002), 607–618.

    Article  MATH  MathSciNet  Google Scholar 

  15. B. Kim: On the derivations of semiprime rings and noncommutative Banach algebras. Acta Math. Sin., Engl. Ser. 16 (2000), 21–28.

    Article  MATH  MathSciNet  Google Scholar 

  16. C. Lanski: Differential identities, Lie ideals, and Posner’s theorems. Pac. J. Math. 134 (1988), 275–297.

    MATH  MathSciNet  Google Scholar 

  17. C. Lanski, S. Montgomery: Lie structure of prime rings of characteristic 2. Pac. J. Math. 42 (1972), 117–136.

    Article  MATH  MathSciNet  Google Scholar 

  18. P. H. Lee, T. K. Lee: Lie ideals of prime rings with derivations. Bull. Inst. Math., Acad. Sin. 11 (1983), 75–80.

    MATH  Google Scholar 

  19. T.-K. Lee: Generalized derivations of left faithful rings. Commun. Algebra 27 (1999), 4057–4073.

    Article  MATH  Google Scholar 

  20. T. K. Lee: Semiprime rings with differential identities. Bull. Inst. Math., Acad. Sin. 20 (1992), 27–38.

    MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  22. M. Mathieu: Properties of the product of two derivations of a C*-algebra. Can. Math. Bull. 32 (1989), 490–497.

    Article  MATH  MathSciNet  Google Scholar 

  23. M. Mathieu, G. J. Murphy: Derivations mapping into the radical. Arch. Math. 57 (1991), 469–474.

    Article  MATH  MathSciNet  Google Scholar 

  24. K.-H. Park: On derivations in noncommutative semiprime rings and Banach algebras. Bull. Korean Math. Soc. 42 (2005), 671–678.

    Article  MATH  MathSciNet  Google Scholar 

  25. E. C. Posner: Derivations in prime rings. Proc. Am. Math. Soc. 8 (1957), 1093–1100.

    Article  MathSciNet  Google Scholar 

  26. A. M. Sinclair: Continuous derivations on Banach algebras. Proc. Am. Math. Soc. 20 (1969), 166–170.

    Article  MATH  MathSciNet  Google Scholar 

  27. I. M. Singer, J. Wermer: Derivations on commutative normed algebras. Math. Ann. 129 (1955), 260–264.

    Article  MATH  MathSciNet  Google Scholar 

  28. M. P. Thomas: The image of a derivation is contained in the radical. Ann. Math. (2) 128 (1988), 435–460.

    Article  MATH  Google Scholar 

  29. J. Vukman: On derivations in prime rings and Banach algebras. Proc. Am. Math. Soc. 116 (1992), 877–884.

    Article  MATH  MathSciNet  Google Scholar 

  30. B. Yood: Continuous homomorphisms and derivations on Banach algebras. Proceedings of the Conference on Banach Algebras and Several Complex Variables, New Haven, Conn., 1983 (F. Greenleaf et al., eds.). Contemp. Math. 32, Amer. Math. Soc., Providence, 1984, pp. 279–284.

    Chapter  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Belda College, Belda, Paschim Medinipur, West Bengal, 721 424, India

    Basudeb Dhara

  2. Department of Mathematics, Jadavpur University, 188, Raja S.C. Mallick Road, Jadavpur, Kolkata, 700 032, India

    Sukhendu Kar & Sachhidananda Mondal

Authors
  1. Basudeb Dhara
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sukhendu Kar
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Sachhidananda Mondal
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Basudeb Dhara.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Dhara, B., Kar, S. & Mondal, S. Generalized derivations on Lie ideals in prime rings. Czech Math J 65, 179–190 (2015). https://doi.org/10.1007/s10587-015-0167-4

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10587-015-0167-4

Keywords

MSC 2010

Search

Navigation