Let R be a prime ring with char R ≠ 2 and let d be a generalized derivation on R. We study the generalized derivation d satisfying any of the following identities:
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(i)
d[(x, y)] = [d(x), d(y)] for all x , y ∈ R ;
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(ii)
d[(x, y)] = [d(y), d(x)] for all x , y ∈ R ;
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(iii)
either d([x, y]) = [d(x), d(y)] or d([x, y]) = [d(y), d(x)] for all x , y ∈ R .
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References
E. Albaş, N. Argaç, and V. De Filipps, “Generalized derivations with Engel conditions on one-sided ideals,” Commun. Algebra, 36, No. 6, 2063–2071 (2008).
E. Albaş and N. Argaç, “Generalized derivations of prime rings,” Algebra Colloq., 11, No. 3, 399–410 (2004).
M. Ashraf, A. Ali, and S. Ali, “Some commutativity theorems for rings with generalized derivations,” Southeast Asian Bull. Math., 31, 415–421 (2007).
M. Ashraf, A. Ali, and R. Rani, “On generalized derivations of prime rings,” Southeast Asian Bull. Math., 29, No. 4, 669–675 (2005).
K. I. Beidar, “Rings of quotients of semiprime rings,” Vestn. Mosk. Univ., Ser. I., Mekh., 33, 36–42 (1978), English translation: Transl. Moscow Univ. Math. Bull., 33, 29–34 (1978).
K. I. Beidar, W. S. Martindale, and V. Mikhalev, “Rings with generalized identities,” Pure Appl. Math. (1996).
H. E. Bell and N. Rehman, “Generalized derivations with commutativity and anticommutativity conditions,” Math. J. Okayama Univ., 49, 139–147 (2007).
M. Brešar, “On the distance of the composition of two derivations to the generalized derivations,” Glasgow Math. J., 33, 89–93 (1991).
C. L. Chuang, “GPIs having coefficients in Utumi quotient rings,” Proc. Amer. Math. Soc., 103, No. 3, 723–728 (1988).
B. Hvala, “Generalized derivations in prime rings,” Commun. Algebra, 26, No. 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, “Prime rings satisfying a generalized polynomial identity,” J. Algebra, 12, 579–584 (1969).
E. C. Posner, “Derivations in prime rings,” Proc. Amer. Math. Soc., 8, 1093–1100 (1957).
H. Shuliang, “Generalized derivations of prime rings,” Int. J. Math. Math. Sci., 2007, 1–6 (2007).
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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 63, No. 5, pp. 596–602, May, 2011.
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Albaş, E. On generalized derivations satisfying certain identities. Ukr Math J 63, 690–698 (2011). https://doi.org/10.1007/s11253-011-0535-7
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DOI: https://doi.org/10.1007/s11253-011-0535-7