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)
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)
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.
Similar content being viewed by others
References
Argaç, N., Nakajima, A., ALBAŞ, E.: On orthogonal generalized derivations of semiprime rings. Turk. J. Math. 28(2), 185–194 (2004)
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)
Bergen, J., Herstein, I.N., Kerr, J.W.: Lie ideals and derivations of prime rings. J. Algebra 71(1), 259–267 (1981)
Bresar, M.: Orthogonal derivations and extension of a theorem of posner. Radovi Math. 5, 237–246 (1989)
Chuang, C.L.: The additive subgroup generated by a polynomial. Isr. J. Math. 59(1), 98–106 (1987)
Chuang, C.L.: Gpis having coefficients in utumi quotient rings. Proc. Am. Math. Soc. 103(3), 723–728 (1988)
De Filippis, V.: On the annihilator of commutators with derivation in prime rings. Rend. del Circ. Mat. di Palermo 49(2), 343–352 (2000)
De Filippis, V., Di Vincenzo, O.: Posner’s second theorem and an annihilator condition. Math. Pannon. 12(1), 69–81 (2001)
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)
De Filippis, V., Di Vincenzo, O.M.: Vanishing derivations and centralizers of generalized derivations on multilinear polynomials. Commun. Algebra 40(6), 1918–1932 (2012)
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)
Dhara, B., De Filippis, V.: Co-commutators with generalized derivations in prime and semiprime rings. Publ. Math. Debr. 85(3–4), 339–360 (2014)
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)
Erickson, T.S., Martindale 3rd, W.S., Osborn, J.M.: Prime nonassociative algebras. Pac. J. Math. 60(1), 49–63 (1975)
Faith, C., Utumi, Y.: On a new proof of Litoff’s theorem. Acta Math. Acad. Sci. Hung. 14, 369–371 (1963)
Filippis, V.D., Vincenzo, O.M.D.: Vanishing derivations and centralizers of generalized derivations on multilinear polynomials. Commun. Algebra 40(6), 1918–1932 (2012)
Hvala, B.: Generalized derivations in rings. Commun. Algebra 26(4), 1147–1166 (1998)
Jacobson, N.: Structure of Rings. American Mathematical Society Colloquium Publications, Vol.37. Revised Edition. American Mathematical Society, Providence (1964)
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)
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)
Kanel-Belov, A., Malev, S., Rowen, L.: The images of Lie polynomials evaluated on matrices. Commun. Algebra 45(11), 4801–4808 (2017)
Kharchenko, V.K.: Differential identities of prime rings. Algebra Logic 17(2), 155–168 (1978)
Lee, T.K.: Semiprime rings with differential identities. Bull. Inst. Math. Acad. Sin. 20(1), 27–38 (1992)
Lee, T.K.: Generalized derivations of left faithful rings. Commun. Algebra 27(8), 4057–4073 (1999)
Leron, U.: Nil and power-central polynomials in rings. Trans. Am. Math. Soc. 202, 97–103 (1975)
Martindale III, W.S.: Prime rings satisfying a generalized polynomial identity. J. Algebra 12, 576–584 (1969)
Oukhtite, L.: Posner’s second theorem for jordan ideals in rings with involution. Expo. Math. 29(4), 415–419 (2011)
Posner, E.C.: Derivations in prime rings. Proc. Am. Math. Soc. 8(6), 1093–1100 (1957)
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)
Wu, W., Niu, F.W.: Annihilator on co-commutators with derivations on lie ideals in prime rings. Northeast Math. 22(4), 415–424 (2006)
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
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). https://doi.org/10.1007/s12215-019-00473-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12215-019-00473-6