Abstract
Let R be a 6-torsion-free prime ring and let \({D : R \rightarrow R}\) be an additive mapping satisfying the relation 2D(x 4) = D(x 3)x + x 3 D(x) + D(x)x 3 + xD(x 3) for all \({x \in R}\) . The purpose of this paper is to show that D is a derivation. This result is related to a classical result of Herstein, which states that any Jordan derivation on a 2-torsion-free prime ring is a derivation.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Beidar K.I., Fong Y.: On additive isomorphisms of prime rings preserving polynomials. J. Algebra 217(2), 650–667 (1999)
Beidar, K.I., Martindale III, W.S., Mikhalev, A.V.: Rings with Generalized Identities. Marcel Dekker, Inc., New York (1996)
Brešar M.: Jordan derivations on semiprime rings. Proc. Am. Math. Soc 104, 1003–1006 (1988)
Brešar M.: Jordan mappings of semiprime rings. J. Algebra 127, 218–228 (1989)
Brešar M.: Functional identities: a survey. Contemp. Math. 259, 93–109 (2000)
Brešar, M., Chebotar, M., Martindale 3rd, W.S.: Functional Identities. Birkhäuser Verlag, Basel (2007)
Brešar M., Vukman J.: Jordan derivations on prime rings. Bull. Aust. Math. Soc 37, 321–323 (1988)
Cusack J.: Jordan derivations on rings. Proc. Am. Math. Soc. 53, 321–324 (1975)
Fošner A., Fošner M., Vukman J.: Identities with derivations in rings. Glas. Mat. Ser. III 46, 339–349 (2011)
Fošner A., Vukman J.: On certain functional equations related to Jordan triple \({(\theta, \phi)}\) -derivations on semiprime rings. Monatsh. Math. 162, 157–165 (2011)
Fošner A., Vukman J.: Some results concerning additive mappings and derivations on semiprime rings. Publ. Math. Debrecen 78, 575–581 (2011)
Herstein I.N.: Jordan derivations in prime rings. Proc. Am. Math. Soc 8, 1104–1110 (1957)
Vukman J.: Some remarks on derivations in semiprime rings and standard operator algebras. Glasnik Mat. Vol. 46, 43–48 (2011)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Fošner, M., Marcen, B. & Širovnik, N. On Certain Identity Related to Herstein Theorem on Jordan Derivations. Mediterr. J. Math. 13, 537–556 (2016). https://doi.org/10.1007/s00009-014-0512-0
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00009-014-0512-0