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Composition and orthogonality of derivations with multilinear polynomials in prime rings

  • Balchand PrajapatiEmail author
  • Charu Gupta
Article
  • 4 Downloads

Abstract

Let R be a non commutative prime ring of characteristic different from 2 with Utumi quotient ring U and extended centroid C. Let d and \(\delta \) be two derivations of R and S be the set of evaluations of a multilinear polynomial \(f(x_1,\ldots ,x_n)\) over C which is not central valued. Let \(p,q\in R\). We prove the followings.
  1. (1)

    If \(pud\delta (u)+\delta d(u)uq=0\) for all \(u\in S\) and \(p+q\notin C\). Then either \(d=0\) or \(\delta =0\).

     
  2. (2)

    If \(pud(u)+d(u)uq=0\) for all \(u\in S\). Then either \(d=0\) or \(p=q\in C\), \(d(x)=[a,x]\) for some \(a\in U\) and \(f(x_1,\ldots ,x_n)^2\) is central valued.

     

Keywords

Prime ring Derivation Orthogonal derivations Extended centroid Utumi quotient ring 

Mathematics Subject Classification

16W25 16N60 

Notes

Acknowledgements

The authors are highly thankful to the referee for his/her several useful suggestions. First author is partially supported by the research Grant DST-SERB EMR/2016/001550.

References

  1. 1.
    Argaç, N., Nakajima, A., ALBAŞ, E.: On orthogonal generalized derivations of semiprime rings. Turk. J. Math. 28(2), 185–194 (2004)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Beidar, K.I., Martindale III, W.S., Mikhalev, A.V.: Rings with Generalized Identities, vol. 196 of Monographs and Textbooks in Pure and Applied Mathematics. Marcel Dekker, Inc., New York (1996)Google Scholar
  3. 3.
    Bergen, J., Herstein, I.N., Kerr, J.W.: Lie ideals and derivations of prime rings. J. Algebra 71(1), 259–267 (1981)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bresar, M.: Orthogonal derivations and extension of a theorem of posner. Radovi Math. 5, 237–246 (1989)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Chuang, C.L.: The additive subgroup generated by a polynomial. Isr. J. Math. 59(1), 98–106 (1987)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Chuang, C.L.: Gpis having coefficients in utumi quotient rings. Proc. Am. Math. Soc. 103(3), 723–728 (1988)MathSciNetCrossRefGoogle Scholar
  7. 7.
    De Filippis, V.: On the annihilator of commutators with derivation in prime rings. Rend. del Circ. Mat. di Palermo 49(2), 343–352 (2000)MathSciNetCrossRefGoogle Scholar
  8. 8.
    De Filippis, V., Di Vincenzo, O.: Posner’s second theorem and an annihilator condition. Math. Pannon. 12(1), 69–81 (2001)MathSciNetzbMATHGoogle Scholar
  9. 9.
    De Filippis, V., Di Vincenzo, O.M.: Posner’s second theorem, multilinear polynomials and vanishing derivations. J. Aust. Math. Soc. 76(3), 357–368 (2004)MathSciNetCrossRefGoogle Scholar
  10. 10.
    De Filippis, V., Di Vincenzo, O.M.: Vanishing derivations and centralizers of generalized derivations on multilinear polynomials. Commun. Algebra 40(6), 1918–1932 (2012)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Dhara, B., Argac, N., Albas, E.: Vanishing derivations and co-centralizing generalized derivations on multilinear polynomials in prime rings. Commun. Algebra 44(5), 1905–1923 (2016)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Dhara, B., De Filippis, V.: Co-commutators with generalized derivations in prime and semiprime rings. Publ. Math. Debr. 85(3–4), 339–360 (2014)MathSciNetCrossRefGoogle Scholar
  13. 13.
    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)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Erickson, T.S., Martindale 3rd, W.S., Osborn, J.M.: Prime nonassociative algebras. Pac. J. Math. 60(1), 49–63 (1975)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Faith, C., Utumi, Y.: On a new proof of Litoff’s theorem. Acta Math. Acad. Sci. Hung. 14, 369–371 (1963)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Filippis, V.D., Vincenzo, O.M.D.: Vanishing derivations and centralizers of generalized derivations on multilinear polynomials. Commun. Algebra 40(6), 1918–1932 (2012)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Hvala, B.: Generalized derivations in rings. Commun. Algebra 26(4), 1147–1166 (1998)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Jacobson, N.: Structure of Rings. American Mathematical Society Colloquium Publications, Vol.37. Revised Edition. American Mathematical Society, Providence (1964)Google Scholar
  19. 19.
    Kanel-Belov, A., Malev, S., Rowen, L.: The images of non-commutative polynomials evaluated on \(2\times 2\) matrices. Proc. Am. Math. Soc. 140(2), 465–478 (2012)CrossRefGoogle Scholar
  20. 20.
    Kanel-Belov, A., Malev, S., Rowen, L.: The images of multilinear polynomials evaluated on \(3\times 3\) matrices. Proc. Am. Math. Soc. 144(1), 7–19 (2016)CrossRefGoogle Scholar
  21. 21.
    Kanel-Belov, A., Malev, S., Rowen, L.: The images of Lie polynomials evaluated on matrices. Commun. Algebra 45(11), 4801–4808 (2017)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Kharchenko, V.K.: Differential identities of prime rings. Algebra Logic 17(2), 155–168 (1978)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Lee, T.K.: Semiprime rings with differential identities. Bull. Inst. Math. Acad. Sin. 20(1), 27–38 (1992)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Lee, T.K.: Generalized derivations of left faithful rings. Commun. Algebra 27(8), 4057–4073 (1999)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Leron, U.: Nil and power-central polynomials in rings. Trans. Am. Math. Soc. 202, 97–103 (1975)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Martindale III, W.S.: Prime rings satisfying a generalized polynomial identity. J. Algebra 12, 576–584 (1969)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Oukhtite, L.: Posner’s second theorem for jordan ideals in rings with involution. Expo. Math. 29(4), 415–419 (2011)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Posner, E.C.: Derivations in prime rings. Proc. Am. Math. Soc. 8(6), 1093–1100 (1957)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Tiwari, S.K., Sharma, R.K.: Derivations vanishing identities involving generalized derivations and multilinear polynomial in prime rings. Mediterr. J. Math. 14(5), 207 (2017)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Wu, W., Niu, F.W.: Annihilator on co-commutators with derivations on lie ideals in prime rings. Northeast Math. 22(4), 415–424 (2006)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Italia S.r.l., part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Liberal StudiesAmbedkar UniversityDelhiIndia

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