Composition and orthogonality of derivations with multilinear polynomials in prime rings


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.

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  1. 1.

    Argaç, N., Nakajima, A., ALBAŞ, E.: On orthogonal generalized derivations of semiprime rings. Turk. J. Math. 28(2), 185–194 (2004)

    MathSciNet  MATH  Google 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)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Bresar, M.: Orthogonal derivations and extension of a theorem of posner. Radovi Math. 5, 237–246 (1989)

    MathSciNet  MATH  Google Scholar 

  5. 5.

    Chuang, C.L.: The additive subgroup generated by a polynomial. Isr. J. Math. 59(1), 98–106 (1987)

    MathSciNet  Article  Google Scholar 

  6. 6.

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

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google Scholar 

  8. 8.

    De Filippis, V., Di Vincenzo, O.: Posner’s second theorem and an annihilator condition. Math. Pannon. 12(1), 69–81 (2001)

    MathSciNet  MATH  Google 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)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Erickson, T.S., Martindale 3rd, W.S., Osborn, J.M.: Prime nonassociative algebras. Pac. J. Math. 60(1), 49–63 (1975)

    MathSciNet  Article  Google Scholar 

  15. 15.

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

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google Scholar 

  17. 17.

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

    MathSciNet  Article  Google 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)

    Article  Google 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)

    Article  Google 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)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Kharchenko, V.K.: Differential identities of prime rings. Algebra Logic 17(2), 155–168 (1978)

    MathSciNet  Article  Google Scholar 

  23. 23.

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

    MathSciNet  MATH  Google Scholar 

  24. 24.

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

    MathSciNet  Article  Google Scholar 

  25. 25.

    Leron, U.: Nil and power-central polynomials in rings. Trans. Am. Math. Soc. 202, 97–103 (1975)

    MathSciNet  Article  Google Scholar 

  26. 26.

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

    MathSciNet  Article  Google Scholar 

  27. 27.

    Oukhtite, L.: Posner’s second theorem for jordan ideals in rings with involution. Expo. Math. 29(4), 415–419 (2011)

    MathSciNet  Article  Google Scholar 

  28. 28.

    Posner, E.C.: Derivations in prime rings. Proc. Am. Math. Soc. 8(6), 1093–1100 (1957)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google 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)

    MathSciNet  MATH  Google Scholar 

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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.

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Correspondence to Balchand Prajapati.

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Prajapati, B., Gupta, C. Composition and orthogonality of derivations with multilinear polynomials in prime rings. Rend. Circ. Mat. Palermo, II. Ser 69, 1279–1294 (2020).

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  • Prime ring
  • Derivation
  • Orthogonal derivations
  • Extended centroid
  • Utumi quotient ring

Mathematics Subject Classification

  • 16W25
  • 16N60