JORDAN TRIPLE (α, β) ∗ -DERIVATIONS ON SEMIPRIME RINGS WITH INVOLUTION

Let R be a 2-torsion free semiprime ∗-ring. The aim of this paper isto show that every Jordan triple (α, β)∗ -derivation on R is a Jordan(α, β)∗ -derivation. Furthermore, every Jordan triple left α∗ -centralizeron R is a Jordan left α∗ -centralizer. Consequently, every generalizedJordan triple (α, β)∗ -derivation on R is a Jordan (α, β)∗ -derivation.

Keywords:

ring, ∗-ring, (α, β)∗ - derivation, Jordan (α, β)∗ -derivation, Jordantriple (α, β)∗ -derivation, generalized Jordan triple (α β)∗ -derivation.,

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Ali, Shakir and Foˇ sner, A., On Jordan (α, β) ∗ -derivations in semiprime ∗-rings, International J. Algebra 4, 99–108, 2010.
Breˇ sar M., On the distance of the compositions of two derivations to the generalized derivations, Glasgow Math. J. 33 No.1, 89–93, 1991.
Breˇ sar M., Jordan derivations on semiprime rings, Proc. Amer. Math. Soc. 104, 1003–1006, 198
Breˇ sar M., Jordan mappings of semiprime rings, J. Algebra 127, 218–228, 1989.
Breˇ sar M. and Vukman J., On some additive mappings in rings with involution, Aequ. Math. 38, 178–185, 1989.
Breˇ sar M. and Vukman J., Jordan derivations on prime rings, Bull. Austral. Math. Soc. 37, 321–322, 1988.
Breˇ sar, M. and Zalar, B., On the structure of Jordan ∗-derivation, Colloq. Math. 63, 163– 171, 1992.
Cusack J. M., Jordan derivations on rings. Proc. Amer. Math. Soc. 53, 321–324, 1975.
Daif, M. N. and Tammam El-Sayiad, M. S., On Jordan and Jordan ∗-generalized derivations in semiprime rings with involution, Int. J. Contemp. Math. Scie. 2, 1487–1492, 2007.
Foˇ sner, M. and Iliˇ sevi´ c, D., On Jordan triple derivations and related mappings, Mediterr. J. Math 5, 415–425, 2008.
Foˇ sner, M. and Iliˇ sevi´ c, D., On a class of projections on ∗-rings, Comm. Algebra 33, 3293– 3310, 2005.
Herstein I. N., Topics in Ring Theory, (Chicago Univ Press, Chicago, 1969).
Hongan, M., Rehman, N. and Al-Omary, R. M., Lie ideals and Jordan triple derivations in rings, Rend. Sem. Mat. Univ. Padova, 125, 147–156, 2011.
Iliˇ sevi´ c , D., Quadratic functionals on modules over ∗-rings, Studia Sci. Math. Hungar. 42, 95–105, 2005.
Liu, C. K. and Shiue, Q. K., Generalized Jordan triple (θ, φ)-derivations of semiprime rings, Taiwanese J. Math. 11, 1397–1406, 2007.
Najati, Abbas, Jordan θ-derivation on Lie triple systems, Bull. Korean Math. Soc. 46 No. 3, 435–437, 2009.
Najati, Abbas, On generalized Jordan derivations of Lie triple systems, Czechoslovak Mathematical Journal, 60 No. 135, 541–547, 2010.
ˇ Semrl, P., Quadratic functionls and Jordan ∗-derivations, Studia Math. 97, 157–165, 1991. Hvala B., Generalized derivations in rings, Comm. Algebra 26, 1149–1166, 1998.
Jing W. and Lu S., Generalized Jordan derivations on prime rings and standard operator algebras, Taiwanese J. Math. 7, 605–613, 2003.
Lanski C., Generalized derivations and nth power mappings in rings, Comm Algebra 35, 3660–3672, 2007.
Molnar L., On centralizers of an H ∗ -algebra, Publ. Math. Debrecen 46 No. 1-2, 89–95, 1995. Vukman, J., A note on Jordan ∗-derivations in semiprime rings with involution, Int. math. Forum 13, 617–622, 2006.
Vukman J., A note on generalized derivations of semiprime rings, Taiwanese J. Math. 11, 367–370, 2007.
Zalar B., On centralizers of semiprime rings, Comment. Math. Univ. Carolinae 32, 609–614, 199