Glasnik Matematicki, Vol. 53, No. 1 (2018), 73-95.

A RESULT IN THE SPIRIT OF HERSTEIN THEOREM

Maja Fošner, Benjamin Marcen and Joso Vukman

Faculty of logistics, University of Maribor, Mariborska cesta 7, 3000 Celje, Slovenia
e-mail: maja.fosner@um.si

Faculty of logistics, University of Maribor, Mariborska cesta 7, 3000 Celje, Slovenia
e-mail: benjamin.marcen@um.si

Institute of mathematics, physics and mechanics, Jadranska 19, 1000 Ljubljana, Slovenia
e-mail: joso.vukman@guest.um.si


Abstract.   A classical result of Herstein asserts that any Jordan derivation on a prime ring of characteristic different from two is a derivation. It is our aim in this paper to prove the following result, which is in the spirit of Herstein's theorem. Let n≥ 3 be some fixed integer, let R be a prime ring with char(R)> 4n-8 and let D:R → R be an additive mapping satisfying either the relation D(xn)=D(xn-1)x+xn-1D(x) or the relation D(xn)=D(x)xn-1+xD(xn-1) for all x R. In both cases D is a derivation.

2010 Mathematics Subject Classification.   16W10, 39B05.

Key words and phrases.   Prime ring, semiprime ring, derivation, Jordan derivation, functional identity.


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DOI: 10.3336/gm.53.1.06


References:

  1. K. I. Beidar, M. Brešar, M. A. Chebotar and W. S. Martindale 3rd, On Herstein's Lie map conjectures II, J. Algebra 238 (2001), 239-264.
    MathSciNet     CrossRef

  2. K. I. Beidar and Y. Fong, On additive isomorphisms of prime rings preserving polynomials, J. Algebra 217 (1999), 650-667.
    MathSciNet     CrossRef

  3. M. Brešar, Jordan derivations on semiprime rings, Proc. Amer. Math. Soc. 104 (1988), 1003-1006.
    MathSciNet     CrossRef

  4. M. Brešar, Jordan mappings of semiprime rings, J. Algebra 127 (1989), 218-228.
    MathSciNet     CrossRef

  5. M. Brešar, Functional identities: A survey, in: Algebra and its applications, Contemp. Math. 259, Amer. Math. Soc., Providence, 2000, 93-109.

  6. M. Brešar, M. Chebotar and W. S. Martindale 3rd, Functional identities, Birkhäuser Verlag, Basel, 2007.
    MathSciNet    

  7. M. Brešar and J. Vukman, Jordan derivations on prime rings, Bull. Austral. Math. Soc. 37 (1988), 321-322.
    MathSciNet     CrossRef

  8. M. Brešar and J. Vukman, Jordan (θ ,φ )-derivations, Glas. Mat. Ser. III 26(46) (1991), 13-17.
    MathSciNet    

  9. Cheng-Kai Liu and Wen-Kwei Shiue, Generalized Jordan triple (θ ,φ )-derivations on semiprime rings, Taiwanese J. Math. 11 (2007), 1397-1406.
    MathSciNet     CrossRef

  10. J. M. Cusack, Jordan derivations on rings, Proc. Amer. Math. Soc. 53 (1975), 321-324.
    MathSciNet     CrossRef

  11. M. Fošner and N. Peršin, A result concerning derivations in prime rings, Glas. Mat. Ser. III 48(68) (2013), 67-79.
    MathSciNet     CrossRef

  12. I. N. Herstein, Jordan derivations of prime rings, Proc. Amer. Math. Soc. 8 (1957), 1104-1110.
    MathSciNet     CrossRef

  13. L. Rowen, Some results on the center of a ring with polynomial identity, Bull. Amer. Math. Soc. 79 (1973), 219-223.
    MathSciNet     CrossRef

  14. J. Vukman, On derivations of standard operator algebras and semisimple H* -algebras, Studia Sci. Math. Hungar. 44 (2007), 57-63.
    MathSciNet     CrossRef

  15. J. Vukman, Some remarks on derivations in semiprime rings and standard operator algebras, Glas. Mat. Ser. III 46(66) (2011), 43-48.
    MathSciNet     CrossRef

  16. J. Vukman, I. Kosi-Ulbl and D. Eremita, On certain equations in rings, Bull. Austral. Math. Soc. 71 (2005), 53-60.
    MathSciNet     CrossRef

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