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AIMS Mathematics

2017, Volume 2, Issue 4: 580-585. doi: 10.3934/Math.2017.4.580
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Research article

A note on derivations and Jordan ideals of prime rings

  • Department of Mathematics, Punjabi University, Patiala, Punjab-147001, India
  • Received: 14 September 2017 Accepted: 23 October 2017 Published: 01 November 2017

  • MSC : 16W25, 16N60, 16U80

  • Let $F:R\rightarrow R$ be a generalized derivation of a 2-torsion free prime ring $R$ together with a derivation $d.$ In this paper, we show that a nonzero Jordan ideal $J$ of $R$ contains a nonzero ideal of $R$. Further, we use this result to prove that if $F([x, y])\in Z(R)$ for all $x, y\in J, $ then $R$ is commutative. Consequently, it extends a result of Oukhtite, Mamouni and Ashraf.

    Citation: Gurninder S. Sandhu, Deepak Kumar. A note on derivations and Jordan ideals of prime rings[J]. AIMS Mathematics, 2017, 2(4): 580-585. doi: 10.3934/Math.2017.4.580

    Related Papers:

  • Let $F:R\rightarrow R$ be a generalized derivation of a 2-torsion free prime ring $R$ together with a derivation $d.$ In this paper, we show that a nonzero Jordan ideal $J$ of $R$ contains a nonzero ideal of $R$. Further, we use this result to prove that if $F([x, y])\in Z(R)$ for all $x, y\in J, $ then $R$ is commutative. Consequently, it extends a result of Oukhtite, Mamouni and Ashraf.


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    [1] S. M. A. Zaidi, M. Ashraf, S. Ali, On Jordan ideals and left (θ; θ)-derivations in prime rings. Internat. J. Math. Math. Sci., 37 (2004), 1957-1964.
    [2] L. Oukhtite, Posner's second theorem for Jordan ideals in rings with involutions. Expo. Math., 29 (2011), 415-419.
    [3] L. Oukhtite, A. Mamouni, M. Ashraf, Commutativity theorems for rings with differential identities on Jordan ideals. Comment. Math. Univ. Carolin., 54 (2013), no. 4,447-457.
    [4] L. Oukhtite, On Jordan ideals and derivations in rings with involution. Comment. Math. Univ. Carolin., 51 (2010), no. 3,389-395.
    [5] N. Argacç, V. de Filippis, Actions of generalized derivations on multilinear polynomials in prime rings. Algebra Colloq., 18(spec 1) (2011), 955-964.
    [6] V. K. Kharchenko, Differential identities of prime rings. Algebra Logic, 17 (1979), 155-168.
    [7] C. L. Chuang, GPIs having coeffcients in Utumi quotient rings. Proc. Amer. Math. Soc., 103 (1988), 723-728.
    [8] T. S. Erickson, W. Martindale Ⅲ, J. M. Osborn, Prime nonassociative algebras. Pac. J. Math., 60 (1975), 49-63.
    [9] W. S. Martindale Ⅲ, Prime rings satisfying a generalized polynomial identity. J. Algebra, 12 (1969), 576-584.
    [10] N. Jacobson, Structure of Rings. Colloquium Publications, vol. 37, Amer. Math. Soc. Ⅶ, Provindence, RI, 1956.
    [11] X. W. Xu, The Power Values Properties of Generalized Derivations. Doctoral Thesis of Jilin University, Changchun, 2006.
    [12] K. I. Beidar, W. S. Martindale Ⅲ, A. V. Mikhalev, Rings with Generalized Identities. Pure and Applied Mathematics. vol. 196, Marcel Dekker, New York, 1996.
    [13] M. Brešar, On the distance of the composition of two derivations to the generalized derivations. Glasg. Math. J., 33 (1991), 89-93.
    [14] T. K. Lee, Generalized derivations of left faithful rings. Comm. Algebra, 27 (1999), no. 8,4057-4073.
    [15] R. Awtar, Lie and Jordan structure in prime rings with derivations. Proc. Amer. Math. Soc., 41 (1973), 67-74.
    [16] I. N. Herstein, A note on derivations. Canad. Math. Bull., 21 (1978), 369-370.
    [17] H. E. Bell, M. N. Daif, On derivations and commutativity in prime rings. Acta Math. Hungar., 66 (1995), 337-343.
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  • © 2017 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
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