A note on generalized Lie derivations of prime rings

doi:10.1007/s11464-016-0589-9

Front. Math. China

2017, Vol. 12

Issue (1) : 247-260 https://doi.org/10.1007/s11464-016-0589-9

RESEARCH ARTICLE

A note on generalized Lie derivations of prime rings

Nihan Baydar YARBIL(

),Nurcan ARGAC

Department of Mathematics, Science Faculty, Ege University, 35100 Bornova, Izmir, Turkey

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Abstract

Let R be a prime ring of characteristic not 2, A be an additive subgroup of R, and F, T, D,K: A → R be additive maps such that F[x, y]) = F(x)y − yK(x) − T(y)x + xD(y) for all x, y ∈ A. Our aim is to deal with this functional identity when A is R itself or a noncentral Lie ideal of R. Eventually, we are able to describe the forms of the mappings F, T, D, and K in case A = R with deg(R)>3 and also in the case A is a noncentral Lie ideal and deg(R)>9. These enable us in return to characterize the forms of both generalized Lie derivations, D-Lie derivations and Lie centralizers of R under some mild assumptions. Finally, we give a generalization of Lie homomorphisms on Lie ideals.

Keywords Prime ring derivation generalized derivation generalized Lie derivation functional identity generalized polynomial identity

Corresponding Author(s): Nihan Baydar YARBIL

Issue Date: 17 November 2016

Cite this article:

Nihan Baydar YARBIL,Nurcan ARGAC. A note on generalized Lie derivations of prime rings[J]. Front. Math. China, 2017, 12(1): 247-260.

URL:

https://academic.hep.com.cn/fmc/EN/10.1007/s11464-016-0589-9
https://academic.hep.com.cn/fmc/EN/Y2017/V12/I1/247

1	Albas E. On generalized derivations satisfying certain conditions. Ukrainian Math J, 2011, 63(5): 690–698 https://doi.org/10.1007/s11253-011-0535-7
2	Beidar K I. On functional identities and commuting additive mappings. Comm Algebra, 1998, 26(6): 1819–1850 https://doi.org/10.1080/00927879808826241
3	Beidar K I, Chebotar M A. On functional identities and d-free subsets of rings I. Comm Algebra, 2000, 28(8): 3925–3951 https://doi.org/10.1080/00927870008827066
4	Beidar K I, Chebotar M A. On Lie derivations of Lie ideals of prime algebras. Israel J Math, 2001, 123: 131–148 https://doi.org/10.1007/BF02784122
5	Beidar K I, Martindale W S, Mikhalev A V. Rings with Generalized Identities. Monographs and Textbooks in Pure and Applied Mathematics, Vol 196. New York: Marcel Dekker, Inc, 1996
6	Bergen J, Herstein I N, Kerr J W. Lie ideals and derivations of prime rings. J Algebra, 1981, 71: 259–267 https://doi.org/10.1016/0021-8693(81)90120-4
7	Brěsar M. Commuting traces of biadditive mappings, commutativity-preserving mappings and Lie mappings. Trans Amer Math Soc, 1993, 335: 525–546 https://doi.org/10.1090/S0002-9947-1993-1069746-X
8	Brěsar M. On generalized biderivations and related maps. J Algebra, 1995, 172(3): 764–786 https://doi.org/10.1006/jabr.1995.1069
9	Brěsar M, ˇ Semrl P. Commuting traces of biadditive maps revisited. Comm Algebra, 2003, 31(1): 381–388 https://doi.org/10.1081/AGB-120016765
10	Chuang C L. GPI’s having coefficients in Utumi quotient rings. Proc Amer Math Soc, 1998, 103(3): 723–728 https://doi.org/10.1090/S0002-9939-1988-0947646-4
11	Herstein I N. Topics in Ring Theory. Chicago: Univ of Chicago Press, 1969
12	Hvala B. Generalized derivations in rings. Comm Algebra, 1998, 26(4): 1147–1166 https://doi.org/10.1080/00927879808826190
13	Hvala B. Generalized Lie derivations of prime rings. Taiwanese J Math, 2007, 11(5): 1425–1430
14	Jing W. Additivity of Lie Centralizers on Triangular Rings. Math and Computer Science Working Papers, Paper 8. Fayetteville State Univ, 2011
15	Kharchenko V K. Differential identities of prime rings. Algebra Logic, 1978, 17: 155–168 https://doi.org/10.1007/BF01670115
16	Lee P H, Wong T L. Derivations cocentralizing Lie ideals. Bull Inst Math Acad Sin, 1995, 23(1): 1–5
17	Lee T K. Semiprime rings with differential identities. Bull Inst Math Acad Sin, 1992, 20(1): 27–38
18	Lee T K. Differential identities of Lie ideals or large right ideals in prime rings. Comm Algebra, 1999, 27(2): 793–810 https://doi.org/10.1080/00927879908826462
19	Lee T K. Generalized derivations of left faithful rings. Comm Algebra, 1999, 27(8): 4057–4073 https://doi.org/10.1080/00927879908826682
20	Lee T K, Lin J S. A result on derivations. Proc Amer Math Soc, 1996, 124: 1687–1691 https://doi.org/10.1090/S0002-9939-96-03234-0
21	Liao P B, Liu C K. On generalized Lie derivations of Lie ideals of prime algebras. Linear Algebra Appl, 2009, 430: 1236–1242 https://doi.org/10.1016/j.laa.2008.10.022
22	Nakajima A. On generalized higher derivations. Turkish J Math, 2000, 24: 295–311
23	Posner E C. Prime rings satisfying a polynomial identity. Proc Amer Math Soc, 1960, 11: 180–183 https://doi.org/10.1090/S0002-9939-1960-0111765-5
24	Scudo G. Generalized derivations acting as Lie homomorphisms on polynomials in prime rings. South Asian J Math, 2014, 38(4): 563–572
25	Xu X W, Ma J, Niu F W. Annihilators of power values of generalized derivation. Chinese Ann Math Ser A, 2007, 28: 131–140 (in Chinese)

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