Some differential identities on prime and semiprime rings and Banach algebras

  • Mohd Arif Raza
  • Mohammad Shadab KhanEmail author
  • Nadeem ur Rehman


In this manuscript, we discuss the behaviour and nature of generalized derivation \({\mathscr {G}}\) on a (semi-) prime ring \({\mathscr {R}}\) satisfying certain differential identities over \({\mathscr {I}}\), a nonzero ideal of \({\mathscr {R}}\). Moreover, we extend our purely ring theoretic result to a non-commutative Banach algebras and obtained some range inclusion results of continuous generalized derivations.


Generalized derivation Martindale ring of quotients Prime and semiprime ring Banach algebra 

Mathematics Subject Classification

46J10 16N20 16N60 16W25 


  1. 1.
    Argaç, N., Inceboz, H.G.: Derivation of prime and semiprime rings. J. Korean Math. Soc. 46(5), 997–1005 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Ashraf, M., Rehman, N.: On commutativity of rings with derivations. Results Math. 42(1–2), 3–8 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Beidar, K.I., Martindale III, W.S., Mikhalev, A.V.: Rings with Generalized Identities. Pure and Applied Mathematics, vol. 196. Marcel Dekker, New York (1996)zbMATHGoogle Scholar
  4. 4.
    Bres̆ar, M.: On the distance of the composition of two derivations to the generalized derivations. Glasg. Math. J. 33, 89–93 (1991)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Chuang, C.L.: GPIs having coefficients in Utumi quotient rings. Proc. Am. Math. Soc. 103, 723–728 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Chuang, C.L.: Hypercentral derivations. J. Algebra 166, 34–71 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Erickson, T.S., Martindale III, W.S., Osborn, J.M.: Prime nonassociative algebras. Pac. J. Math. 60, 49–63 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Herstein, I.N.: Center-like elements in prime rings. J. Algebra 60, 567–574 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Huang, S.: Generalized derivations of prime and semiprime rings. Taiwan. J. Math. 16(2), 771–776 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Jacobson, N.: Structure of Rings, vol. 37, 7th edn. American Mathematical Society, Colloquium Publications, Providence (1956)zbMATHGoogle Scholar
  11. 11.
    Johnson, B.E., Sinclair, A.M.: Continuity of derivations and a problem of Kaplansky. Am. J. Math. 90, 1067–1073 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Kharchenko, V.K.: Differential identities of prime rings. Algebra Log. 17, 155–168 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Lanski, C.: An Engel condition with derivation. Proc. Am. Math. Soc. 118, 731–734 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Lee, T.K.: Generalized derivations of left faithful rings. Commun. Algebra 27(8), 4057–4073 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Lee, T.K.: Semiprime rings with differential identities. Bull. Inst. Math. Acad. Sin. 20, 27–38 (1992)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Martindale III, W.S.: Prime rings satisfying a generalized polynomial identity. J. Algebra 12, 576–584 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Mathieu, M., Murphy, G.J.: Derivations mapping into the radical. Arch. Math. 57, 469–474 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Mayne, J.H.: Centralizing mappings of prime rings. Canad. Math. Bull. 27(1), 122–126 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Raza, M.A., Rehman, N.: On generalized derivation in rings and Banach algebras. Kragujev. J. Math. 41(1), 105–120 (2017)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Raza, M.A., Rehman, N.: On prime and semiprime rings with generalized derivations and non-commutative Banach algebras. Proc. Math. Sci. 126(3), 389–398 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Rehman, N., Raza, M.A.: On Lie ideals with generalized derivations and non-commutative Banach algebras. Bull. Malays. Math. Sci. Soc. 40(2), 747–764 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Quadri, M.A., Khan, M.S., Rehman, N.: Generalized derivations and commutativity of prime rings. Indian J. Pure Appl. Math. 34(98), 1393–1396 (2003)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Sinclair, A.M.: Continuous derivations on Banach algebras. Proc. Am. Math. Soc. 20, 166–170 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Singer, I.M., Wermer, J.: Derivations on commutative normed algebras. Math. Ann. 129, 260–264 (1955)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Thomas, M.P.: The image of a derivation is contained in the radical. Ann. Math. 128(2), 435–460 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Xu, X. W.: The power values properties of generalized derivations, Doctoral Thesis of Jilin University, Changchun (2006)Google Scholar

Copyright information

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

Authors and Affiliations

  • Mohd Arif Raza
    • 1
  • Mohammad Shadab Khan
    • 2
    Email author
  • Nadeem ur Rehman
    • 3
  1. 1.Department of Mathematics, Faculty of Science & Arts-RabighKing Abdulaziz UniversityJeddahKingdom of Saudi Arabia
  2. 2.Department of CommerceAligarh Muslim UniversityAligarhIndia
  3. 3.Department of MathematicsAligarh Muslim UniversityAligarhIndia

Personalised recommendations