Let R be a prime ring of characteristic other than 2 and let I be a nonzero right ideal of R. Also let U be the right Utumi quotient ring of R and let C be the center of U. If G is a generalized derivation of R such that [[G(x), x], G(x)] = 0 for all x ∈ I, then R is commutative or there exist a, b ∈ U such that G(x) = ax + xb for all x ∈ R and one of the following assertions is true:
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(1) (a - λ)I = (0) = (b + λ)I for some λ ∈ C,
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(2) (a - λ)I = (0) for some λ ∈ C and b ∈ C.
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References
E. Albaş, N. Argaç, and V. De Filippis, “Generalized derivations with Engel conditions on one-sided ideals,” Commun. Algebra, 36, No. 6, 2063–2071 (2008).
K. I. Beidar, W. S. Martindale, III, and A. V. Mikhalev, Rings with Generalized Identities, Marcel Dekker, New York (1996).
M. Bresar, “One-sided ideals and derivations of prime rings,” Proc. Am. Math. Soc., 122, No. 4, 979–983 (1994).
C. L. Chuang, “GPI’s having coefficients in Utumi quotient rings,” Proc. Am. Math. Soc., 103, No. 3, 723–728 (1988).
B. Felzenszwalb, “Derivations in prime rings,” Proc. Am. Math. Soc., 84, No. 1, 16–20 (1982).
V. De Filippis and M. S. Tammam El-Sayiad, “A note on Posner’s theorem with generalized derivations on Lie ideals,” Rend. Semin. Mat. Univ. Padova, 122, 55–64 (2009).
V. De Filippis, “Generalized derivations in prime rings and noncommutative Banach algebras,” Bull. Korean Math. Soc., 45, No. 4, 621–629 (2008).
J. S. Erickson, W. S. Martindale, III, and J. M. Osborn, “Prime nonassociative algebras,” Pacif. J. Math., 60, 49–63 (1975).
I. N. Herstein, Topics in Ring Theory, Univ. Chicago Press, Chicago (1969).
B. Hvala, “Generalized derivations in rings,” Commun. Algebra, 26(4), 1147–1166 (1998).
V. K. Kharchenko, “Differential identities of prime rings,” Algebra Logic, 17, 155–168 (1978).
T. K. Lee, “Semiprime rings with differential identities,” Bull. Inst. Math. Acad. Sinica, 20, No. 1, 27–38 (1992).
T. K. Lee, “Generalized derivations of left faithful rings,” Commun. Algebra, 27, No. 8, 4057–4073 (1999).
T. K. Lee and W. K. Shiue, “Identities with generalized derivations,” Commun. Algebra, 29, No. 10, 4435–4450 (2001).
W. S. Martindale III, “Prime rings satisfying a generalized polynomial identity,” J. Algebra, 12, 576–584 (1969).
A. Richoux, “A theorem for prime rings,” Proc. Am. Math. Soc., 77, No. 1, 27–31 (1979).
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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 64, No. 2, pp. 165–175, February, 2012.
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Demir, Ç., Argaç, N. A result on generalized derivations on right ideals of prime rings. Ukr Math J 64, 186–197 (2012). https://doi.org/10.1007/s11253-012-0637-x
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DOI: https://doi.org/10.1007/s11253-012-0637-x