Skip to main content
SpringerLink
Log in

A Note on Generalized Derivations of Order 2 and Multilinear Polynomials in Prime Rings

  • Published:
Acta Mathematica Vietnamica Aims and scope Submit manuscript

Abstract

Let R be a prime ring of char(R)≠ 2, U its Utumi ring of quotients and center C = Z(U) its extended centroid, I a both sided ideal of R, f(x1,…,xn) a multilinear polynomial over C, that is noncentral-valued on R, F, G be two generalized derivations of R and d be a derivation of R. Let f(I) be the set of all evaluations of the multilinear polynomial f(x1,…,xn) in I. If @@@ for all u ∈ f(I), then all possible forms of the maps are determined. As an application of this result, we also study the commutator identity [F2(u)u,G2(v)v] = 0 for all u,v ∈ f(I), where F and G are two generalized derivations of 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. Ali, A., De Filippis, V., Shujat, F.: Commuting values of generalized derivations on multilinear polynomials. Comm. Algebra 42(9), 3699–3707 (2014)

    Article  MathSciNet  Google Scholar 

  2. Argac, N., De Filippis, V.: Actions of generalized derivations on multilinear polynomials in prime rings. Algebra Colloq. 18(Spec 01), 955–964 (2011)

    Article  MathSciNet  Google Scholar 

  3. Carini, L., De Filippis, V.: Centralizers of generalized derivations on multilinear polynomials in prime rings. Siberian Math. J. 53(6), 1051–1060 (2012)

    Article  MathSciNet  Google Scholar 

  4. Chuang, C.L.: GPIS having coefficients in Utumi quotient rings. Proc. Am. Math. Soc. 103(3), 723–728 (1988)

    Article  MathSciNet  Google Scholar 

  5. De Filippis, V., Di Vincenzo, O.M.: Vanishing derivations and centralizers of generalized derivations on multilinear polynomials. Comm. Algebra 40(6), 1918–1932 (2012)

    Article  MathSciNet  Google Scholar 

  6. Dhara, B.: Generalized derivations acting on multilinear polynomials in prime rings. Czechoslovak. Math. J. 68(1), 95–119 (2018)

    MathSciNet  MATH  Google Scholar 

  7. Dhara, B., Argac, N.: Generalized derivations acting on multilinear polynomials in prime rings and Banach algebras. Commun. Math. Stat. 4, 39–54 (2016)

    Article  MathSciNet  Google Scholar 

  8. Dhara, B., Raza, M.A., Ur Rehman, N.: Commutator identity involving generalized derivations on multilinear polynomials. Ann. Univ. Ferrara 62, 205–216 (2016)

    Article  MathSciNet  Google Scholar 

  9. Dhara, B., De Filippis, V.: Co-commutators with generalized derivations in prime and semiprime rings. Publ. Math. Debrecen 85(3–4), 339–360 (2014)

    Article  MathSciNet  Google Scholar 

  10. Dhara, B., Sharma, R.K.: Right sided ideals and multilinear polynomials with derivations on prime rings. Rend. Sem. Mat. Univ. Padova 121, 243–257 (2009)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  12. Eroǧlu, M.P., Argaç, N.: On identities with composition of generalized derivations. Canadian Math. Bull. 60(4), 721–735 (2017)

    Article  MathSciNet  Google Scholar 

  13. Faith, C., Utumi, Y.: On a new proof of Litoff’s theorem. Acta. Math. Acad. Sci. Hung. 14, 369–371 (1963)

    Article  MathSciNet  Google Scholar 

  14. Jacobson, N.: Structure of Rings. Am. Math. Soc. Colloq Pub. 37, Am. Math. Soc., Providence, RI (1964)

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

    Article  Google Scholar 

  16. Lee, T.K.: Generalized derivations of left faithful rings. Comm. Algebra 27(8), 4057–4073 (1999)

    Article  MathSciNet  Google Scholar 

  17. Lee, T.K.: Semiprime rings with differential identities. Bull. Inst. Math. Acad. Sinica 20(1), 27–38 (1992)

    MathSciNet  MATH  Google Scholar 

  18. Leron, U.: Nil and power central polynomials in rings. Trans. Am. Math. Soc. 202, 297–103 (1975)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  20. Tiwari, S.K., Sharma, R.K., Dhara, B.: Derivations vanishing on commutators with generalized derivation of order 2 in prime rings. Comm. Algebra 45 (8), 3542–3554 (2017)

    Article  MathSciNet  Google Scholar 

  21. Wong, T.L.: Derivations with power central values on multilinear polynomials. Algebra Colloquium 3(4), 369–378 (1996)

    MathSciNet  MATH  Google Scholar 

  22. Wong, T.L.: Derivations cocentralizing multilinear polynomials. Taiwanese. J. Math. 1, 31–37 (1997)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The third author expresses his thanks to the University Grants Commission, New Delhi, for its JRF awarded to him under UGC-Ref. No.: 1168/(CSIR-UGC NET JUNE 2018) dated 16.04.2019.

Author information

Authors and Affiliations

  1. Department of Mathematics, Belda College, Belda, Paschim Medinipur, 721424, W.B., India

    Basudeb Dhara

  2. Department of Mathematics, Jadavpur University, Kolkata, India

    Sukhendu Kar & Swarup Kuila

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. Swarup Kuila
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Basudeb Dhara.

Ethics declarations

Conflict of Interest

The authors declare no competing interests.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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. & Kuila, S. A Note on Generalized Derivations of Order 2 and Multilinear Polynomials in Prime Rings. Acta Math Vietnam 47, 755–773 (2022). https://doi.org/10.1007/s40306-021-00471-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40306-021-00471-w

Keywords

Mathematics Subject Classification (2020)

Search

Navigation