Skip to main content
SpringerLink
Log in

Generalized Derivations Acting on Multilinear Polynomials as Jordan Homomorphisms

  • Published:
Ukrainian Mathematical Journal Aims and scope

Let R be a prime ring whose characteristic is not equal to 2, let U be the Utumi quotient ring of R, and let C be the extended centroid of R. Also let G and H be two generalized derivations on R and let f(x1, . . . ,xn) be a noncentral multilinear polynomial over C. If G(H(u2)) = (H(u))2 for all u = f(r1, . . . , rn), r1, . . . , rn ∈ R, then one of the following holds:

(i) H = 0;

(ii) there exists λ ∈ C such that G(x) = H(x) = λx for all x ∈ R;

(iii) there exist λ ∈ C and a ∈ U such that H(x) = λx and G(x) = [a, x]+λx for all x ∈ R and f(x1, . . . ,xn)2 is central-valued on R..

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. V. De Filippis, “Generalized derivations as Jordan homomorphisms on Lie ideals and right ideals,” Acta Math. Sin. (Engl. Ser.), 25, No. 12, 1965–1974 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  2. V. De Filippis, “Generalized skew derivations as Jordan homomorphisms on multilinear polynomials,” J. Korean Math. Soc., 52, No. 1, 191–207 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  3. V. De Filippis and G. Scudo, “Generalized derivations which extend the concept of Jordan homomorphism,” Publ. Math. Debrecen, 86, No. 1-2, 187–212 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  4. V. De Filippis and B. Dhara, “Generalized skew-derivations and generalization of homomorphism maps in prime rings,” Comm. Algebra, 47, No. 8, 3154–3169 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  5. B. Dhara, “Generalized derivations acting as a homomorphism or anti-homomorphism in semiprime rings,” Beitr. Algebra Geom., 53, 203–209 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  6. B. Dhara, “Generalized derivations acting on multilinear polynomials in prime rings,” Czechoslovak Math. J., 68, No. 1, 95–119 (2018).

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  8. H. E. Bell and L. C. Kappe, “Rings in which derivations satisfy certain algebraic conditions,” Acta Math. Hungar., 53, 339–346 (1989).

    Article  MathSciNet  MATH  Google Scholar 

  9. L. Carini, V. De Filippis, and G. Scudo, “Identities with product of generalized skew derivations on multilinear polynomials,” Comm. Algebra, 44, No. 7, 3122–3138 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  10. C. L. Chuang, “GPIs having coefficients in Utumi quotient rings,” Proc. Amer. Math. Soc., 103, No. 3, 723–728 (1988).

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  12. C. Faith and Y. Utumi, “On a new proof of Litoff’s theorem,” Acta Math. Acad. Sci. Hungar., 14, 369–371 (1963).

    Article  MathSciNet  MATH  Google Scholar 

  13. I. N. Herstein, “Jordan homomorphisms,” Trans. Amer. Math. Soc., 81, 331–341 (1956).

    Article  MathSciNet  MATH  Google Scholar 

  14. N. Jacobson, “Structure of rings,” Amer. Math. Soc. Colloq. Publ., 37, American Mathematical Society, Providence, R.I. (1964).

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

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  18. U. Leron, “Nil and power central polynomials in rings,” Trans. Amer. Math. Soc., 202, 97–103 (1975).

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  20. M. F. Smiley, “Jordan homomorphisms onto prime rings,” Trans. Amer. Math. Soc., 84, 426–429 (1957).

    Article  MathSciNet  MATH  Google Scholar 

  21. E. Albas and N. Argac, “Generalized derivations of prime rings,” Algebra Colloq., 11, 399–410 (2004).

    MathSciNet  MATH  Google Scholar 

  22. A. Asma, N. Rehman, and A. Shakir, “On Lie ideals with derivations as homomorphisms and anti-homomorphisms,” Acta Math. Hungar., 101, 79–82 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  23. S. K. Tiwari, “Generalized derivations with multilinear polynomials in prime rings,” Comm. Algebra, 46, No. 12, 5356–5372 (2018).

    Article  MathSciNet  MATH  Google Scholar 

  24. S. K. Tiwari, R. K. Sharma, and B. Dhara, “Identities related to generalized derivation on ideal in prime rings,” Beitr. Algebra Geom., 57, No. 4, 809–821 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  25. S. K. Tiwari, R. K. Sharma, and B. Dhara, “Multiplicative (generalized)-derivation in semiprime rings,” Beitr. Algebra Geom., 58, No. 1, 211–225 (2017).

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Indian Institute of Technology Patna, Bihar, India

    S. K. Tiwari

  2. School of Liberal Studies, Ambedkar University of Delhi, Delhi, India

    B. Prajapati

Authors
  1. S. K. Tiwari
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. B. Prajapati
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. K. Tiwari.

Additional information

Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 74, No. 7, pp. 991–1003, July, 2022. Ukrainian DOI: https://doi.org/10.37863/umzh.v74i7.6108.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tiwari, S.K., Prajapati, B. Generalized Derivations Acting on Multilinear Polynomials as Jordan Homomorphisms. Ukr Math J 74, 1134–1148 (2022). https://doi.org/10.1007/s11253-022-02125-y

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11253-022-02125-y

Search

Navigation