Abstract
Let R be a 2-torsion free semiprime *-ring and let α, β be surjective endomorphisms of R. The aim of the paper is to show that every generalized Jordan triple (α, β)*-derivation on R is a generalized Jordan (α, β)*-derivation. This result makes it possible to prove that every generalized Jordan triple (α, β)*-derivation on a semisimple H*- algebra is a generalized Jordan (α, β)*-derivation. Finally, we prove that every Jordan triple left α*-centralizer on a 2-torsion free semiprime ring is a Jordan left α*-centralizer.
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References
S. Ali, A note on Jordan triple (α, β)*-derivation in H*-algebras, East-West J. Math. 13 (no. 2) (2011), 139-146.
Ali S.: On generalized *-derivation in *-rings. Palest. J. Math 1, 32–37 (2012)
Ali S., Fošner A.: On Jordan (α, β)*-derivation in semiprime *-rings. Int. J. Algebra 4, 99–108 (2010)
S. Ali and C. Haetinger, On Jordan α-centralizer on rings and some applications, Bol. Soc. Parna. Mat. 26 (2008), 71–80.
S. Ali, N. A. Dar and J. Vukman, Jordan left *-centralizers of prime and semiprime rings with involution, Beitr. Algebra Geom. (2012). (DOI:10.1007/s13366-012-0117-3.)
Ambrose W.: Structure theorems for a special class of Banach algebras. Trans. Amer. Math. Soc 57, 364–386 (1945)
M. Ashraf, A. Ali and S. Ali, On Lie ideals and generalized \(({\theta, \varphi})\) -derivations in prime rings, Comm. Algebra 32 (2004), 2977–2985.
Brešar M.: Jordan mappings of semiprime rings. J. Algebra 127, 218–228 (1989)
Brešar M.: On distance of the composition of two derivations to the generalized derivations. Glasg. Math. J. 33, 89–93 (1991)
Brešar M., Vukman J.: On some additive mappings in rings with involution. Aequationes Math. 38, 178–185 (1989)
Brešar M., Zalar B.: On the structure of Jordan *-derivation. Colloq. Math. 63, 163–171 (1992)
M. N. Daif and M. S. Tammam El-Sayiad, On Jordan and Jordan *-generalized derivation in semiprime rings with involution, Int. J.Contemp. Math. Sci. 2 (2007), 1487–1492.
Fošner M., Iliševic D.: On Jordan triple derivations and related mappings. Mediterr. J. Math. 5, 415–427 (2008)
Fošner M., Vukman J.: On some equations in prime rings. Monatsh. Math. 152, 135–150 (2007)
Herstein I.N.: Jordan derivations of prime rings. Proc. Amer. Math. Soc. 8, 1104–1110 (1957)
I. N. Herstein, Rings with involution, Chicago Lectures in Mathematics, The University of Chicago Press, Chicago, 1976.
Iliševic D.: Quadratic functionals on modules over *-rings. Studia Sci. Math. Hungar. 42, 95–105 (2005)
Jing W., Lu S.: Generalized Jordan derivations on prime rings and standard operaror algebras. Taiwanese J. Math. 7, 605–613 (2003)
Lanski C.: Generalized Jordan derivations and nth power maps in rings. Comm. Algebra 35, 3660–3672 (2007)
C. K. Liu and W. K. Shiue, Generalized Jordan triple \({(\theta, \phi)}\) -derivations on semiprime rings, Taiwanese J. Math. 35 (2007), 1397–1406.
Šemrl P.: On Jordan *-derivations and an application. Colloq. Math. 59, 241–251 (1990)
Šemrl P.: Quadratic functionals and Jordan *-derivations. Studia Math. 97, 157–165 (1991)
Vukman J.: A note on generalized derivations of semiprime rings. Taiwanese J. Math. 11, 367–370 (2007)
Vukman J.: A note on Jordan *-derivations in semiprime rings with involution. Int. Math. Forum 13, 617–622 (2006)
Zalar B.: Jordan *-derivation pairs and quadratic functionals on modules over *-rings. Aequationes Math. 54, 31–43 (1997)
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The research of the first two authors is partially supported by the Research Grants (UGC No. 39-37/2010(S(R)) and (INT/SLOVENIA/P-18/2009).
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Ali, S., Fošner, A., Fošner, M. et al. On Generalized Jordan Triple (α, β)*-Derivations and Related Mappings. Mediterr. J. Math. 10, 1657–1668 (2013). https://doi.org/10.1007/s00009-013-0277-x
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DOI: https://doi.org/10.1007/s00009-013-0277-x