On Lie ideals with generalized \((\alpha , \alpha )\)-derivations in prime rings



Let R be a prime ring and \(\alpha \) an automorphism on R. An additive mapping F on R is said to be a generalized \((\alpha , \alpha )\)-derivation on R if there exists an \((\alpha , \alpha )\)-derivation d on R such that \(F(xy)=F(x)\alpha (y)+\alpha (x)d(y)\) holds for all \(x, y\in R\). In this paper our main objective is to study the following identities: (i) \(G(xy) \pm F(x)F(y)\in Z(R);\) (ii) \(G(xy)\pm F(x)F(y)\pm \alpha (yx) =0;\) (iii) \(G(xy) \pm F(x)F(y)\pm \alpha (xy)\in Z(R);\) (iv) \(G(xy)\pm F(x)F(y)\pm \alpha ([x, y]) =0;\) (v) \(G(xy)\pm F(x)F(y)\pm \alpha (x\circ y)=0;\) for all xy in some suitable subset of R, where G and F are two generalized \((\alpha ,\alpha )\)-derivations on R


Prime ring Generalized \((\alpha , \alpha )\)-derivation Automorphism Square closed Lie ideal 

Mathematics Subject Classification

16W25 16N60 16U80 



The author are greatly indebted to the referee for his/her several useful suggestions and valuable comments.


  1. 1.
    Albas, E.: Generalized derivations on ideals of prime rings. Miskolc Mathe. Notes 14(1), 3–9 (2013)MathSciNetMATHGoogle Scholar
  2. 2.
    Ali, S., Dhara, B., Fošner, A.: Some commutativity theorems concerning additive mappings and derivations on semiprime rings. In: Kwak, et al. (eds.) Proceedings of 6th China–Japan–Korea Conference, pp. 133–141. World Scientific, Singapore (2011)Google Scholar
  3. 3.
    Ashraf, M., Rehman, N.: On derivations and commutativity in prime rings. East West J. Math. 3(1), 87–91 (2001)MathSciNetMATHGoogle Scholar
  4. 4.
    Ashraf, M., Ali, A., Ali, S.: Some commutativity theorem for prime rings with generalized derivations. Southeast Asian Bull. Math. 31, 415–421 (2007)MathSciNetMATHGoogle Scholar
  5. 5.
    Atteya, M.J.: On generalized derivations of semiprime rings. Int. J. Algebra 4(12), 591–598 (2010)MathSciNetMATHGoogle Scholar
  6. 6.
    Bergen, J., Herstein, I.N., Kerr, J.W.: Lie ideals and derivations of prime rings. J. Algebra 71(1), 259–267 (1981)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Brešar, M.: On the distance of the composition of two derivations to the generalized derivations. Glasgow Math 33, 89–93 (1991)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Dhara, B.: Generalized derivations acting as a homomorphism or anti-homomorphism in semiprime rings. Beitr. Algebra Geom. 53, 203–209 (2012)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Dhara, B., Ali, S.: On multiplicative (generalized)-derivations in prime and semiprime rings. Aequ. Math. 86(1–2), 65–79 (2013)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Marubayashi, H., Ashraf, M., Rehman, N., Ali, S.: On generalized \((\alpha , \beta )\)-derivations in prime rings Algebra Colloq. 17(spec 01), 865–874 (2010)Google Scholar
  11. 11.
    Posner, E.C.: Derivations in prime rings. Proc. Am. Math. Soc. 8(6), 1093–1100 (1957)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Shuliang, H.: Generalized derivations of \(\ast \)-prime rings. Int. J. Algebra 2(18), 867–873 (2008)MathSciNetMATHGoogle Scholar
  13. 13.
    Tiwari, S.K., Sharma, R.K., Dhara, B.: Multiplicative (generalized)-derivation in semiprime rings. Beitr. Algebra Geom. 58(1), 211–225 (2017)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Italia S.r.l., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of Technology DelhiHauz Khas, New DelhiIndia

Personalised recommendations