Previous |  Up |  Next

Article

Title: An identity with generalized derivations on Lie ideals, right ideals and Banach algebras (English)
Author: de Filippis, Vincenzo
Author: Scudo, Giovanni
Author: Tammam El-Sayiad, Mohammad S.
Language: English
Journal: Czechoslovak Mathematical Journal
ISSN: 0011-4642 (print)
ISSN: 1572-9141 (online)
Volume: 62
Issue: 2
Year: 2012
Pages: 453-468
Summary lang: English
.
Category: math
.
Summary: Let $R$ be a prime ring of characteristic different from $2$, $U$ the Utumi quotient ring of $R$, $C=Z(U)$ the extended centroid of $R$, $L$ a non-central Lie ideal of $R$, $F$ a non-zero generalized derivation of $R$. Suppose that $[F(u),u]F(u)=0$ for all $u\in L$, then one of the following holds: (1) there exists $\alpha \in C$ such that $F(x)=\alpha x$ for all $x\in R$; (2) $R$ satisfies the standard identity $s_4$ and there exist $a\in U$ and $\alpha \in C$ such that $F(x)=ax+xa+\alpha x$ for all $x\in R$. We also extend the result to the one-sided case. Finally, as an application we obtain some range inclusion results of continuous or spectrally bounded generalized derivations on Banach algebras. (English)
Keyword: prime rings
Keyword: differential identities
Keyword: generalized derivations
Keyword: Banach algebra
MSC: 16N60
MSC: 16W25
MSC: 47B47
MSC: 47B48
idZBL: Zbl 1249.16045
idMR: MR2990186
DOI: 10.1007/s10587-012-0039-0
.
Date available: 2012-06-08T09:45:01Z
Last updated: 2016-04-07
Stable URL: http://hdl.handle.net/10338.dmlcz/142838
.
Reference: [1] Beidar, K. I.: Rings with generalized identities. III.Mosc. Univ. Math. Bull. 33 (1978), 53-58. Zbl 0407.16002, MR 0510966
Reference: [2] Beidar, K. I., III, W. S. Martindale, Mikhalev, A. V.: Rings with Generalized Identities.Pure and Applied Mathematics, Marcel Dekker. 196. New York (1996). MR 1368853
Reference: [3] Brešar, M., Mathieu, M.: Derivations mapping into the radical III.J. Funct. Anal. 133 (1995), 21-29. Zbl 0897.46045, MR 1351640, 10.1006/jfan.1995.1116
Reference: [4] Chuang, C. L.: GPIs having coefficients in Utumi quotient rings.Proc. Am. Math. Soc. 103 (1988), 723-728. Zbl 0656.16006, MR 0947646, 10.1090/S0002-9939-1988-0947646-4
Reference: [5] Filippis, V. De: On the annihilator of commutators with derivation in prime rings.Rend. Circ. Mat. Palermo, II. Ser. 49 (2000), 343-352. Zbl 0962.16017, MR 1765404
Reference: [6] Filippis, V. De: A result on vanishing derivations for commutators on right ideals.Math. Pannonica 16 (2005), 3-18. Zbl 1081.16036, MR 2134234
Reference: [7] Filippis, V. De: A product of generalized derivations on polynomials in prime rings.Collect. Math. 61 (2010), 303-322. MR 2732374, 10.1007/BF03191235
Reference: [8] Vincenzo, O. M. Di: On the n-th centralizer of a Lie ideal.Boll. Unione Mat. Ital., VII. Ser., A 3 (1989), 77-85. Zbl 0692.16022, MR 0990089
Reference: [9] Erickson, T. S., III, W. S. Martindale, Osborn, J. M.: Prime nonassociative algebras.Pac. J. Math. 60 (1975), 49-63. MR 0382379, 10.2140/pjm.1975.60.49
Reference: [10] Faith, C., Utumi, Y.: On a new proof of Litoff's theorem.Acta Math. Acad. Sci. Hung. 14 (1963), 369-371. Zbl 0147.28602, MR 0155858, 10.1007/BF01895723
Reference: [11] Herstein, I. N.: Topics in Ring Theory.Chicago Lectures in Mathematics. Chicago-London: The University of Chicago Press. XI (1969). Zbl 0232.16001, MR 0271135
Reference: [12] Jacobson, N.: PI-Algebras. An Introduction.Lecture Notes in Mathematics. 441. Springer-Verlag, New York (1975). Zbl 0326.16013, MR 0369421
Reference: [13] Jacobson, N.: Structure of Rings.Amererican Mathematical Society. Providence R.I. (1956). Zbl 0073.02002, MR 0081264
Reference: [14] Johnson, B. E., Sinclair, A. M.: Continuity of derivations and a problem of Kaplansky.Am. J. Math. 90 (1968), 1067-1073. Zbl 0179.18103, MR 0239419, 10.2307/2373290
Reference: [15] Kharchenko, V. K.: Differential identities of prime rings.Algebra Logic 17 (1979), 155-168. MR 0541758, 10.1007/BF01670115
Reference: [16] Kim, B.: On the derivations of semiprime rings and noncommutative Banach algebras.Acta Math. Sin., Engl. Ser. 16 (2000), 21-28. Zbl 0973.16020, MR 1760520, 10.1007/s101149900020
Reference: [17] Kim, B.: Derivations of semiprime rings and noncommutative Banach algebras.Commun. Korean Math. Soc. 17 (2002), 607-618. Zbl 1101.46317, MR 1971004, 10.4134/CKMS.2002.17.4.607
Reference: [18] Lanski, C.: Differential identities, Lie ideals, and Posner's theorems.Pac. J. Math. 134 (1988), 275-297. Zbl 0614.16028, MR 0961236, 10.2140/pjm.1988.134.275
Reference: [19] Lee, T.-K.: Generalized derivations of left faithful rings.Commun. Algebra 27 (1999), 4057-4073. Zbl 0946.16026, MR 1700189, 10.1080/00927879908826682
Reference: [20] Lee, T.-K.: Semiprime rings with differential identities.Bull. Inst. Math., Acad. Sin. 20 (1992), 27-38. Zbl 0769.16017, MR 1166215
Reference: [21] III, W. S. Martindale: Prime rings satisfying a generalized polynomial identity.J. Algebra 12 (1969), 576-584. MR 0238897, 10.1016/0021-8693(69)90029-5
Reference: [22] Mathieu, M., Murphy, G. J.: Derivations mapping into the radical.Arch. Math. 57 (1991), 469-474. Zbl 0714.46038, MR 1129522, 10.1007/BF01246745
Reference: [23] Mathieu, M., Runde, V.: Derivations mapping into the radical II.Bull. Lond. Math. Soc. 24 (1992), 485-487. Zbl 0733.46023, MR 1173946, 10.1112/blms/24.5.485
Reference: [24] Park, K.-H.: On derivations in noncommutative semiprime rings and Banach algebras.Bull. Korean Math. Soc. 42 (2005), 671-678. Zbl 1105.16031, MR 2181155, 10.4134/BKMS.2005.42.4.671
Reference: [25] Posner, E. C.: Derivations in prime rings.Proc. Am. Math. Soc. 8 (1958), 1093-1100. Zbl 0082.03003, MR 0095863, 10.1090/S0002-9939-1957-0095863-0
Reference: [26] Sinclair, A. M.: Continuous derivations on Banach algebras.Proc. Am. Math. Soc. 20 (1969), 166-170. Zbl 0164.44603, MR 0233207, 10.1090/S0002-9939-1969-0233207-X
Reference: [27] Singer, I. M., Wermer, J.: Derivations on commutative normed algebras.Math. Ann. 129 (1955), 260-264. Zbl 0067.35101, MR 0070061, 10.1007/BF01362370
Reference: [28] Thomas, M. P.: The image of a derivation is contained in the radical.Ann. Math. (2) 128/3 (1988), 435-460. Zbl 0681.47016, MR 0970607
Reference: [29] Wong, T. L.: Derivations with power-central values on multilinear polynomials.Algebra Colloq. 3 (1996), 369-378. Zbl 0864.16031, MR 1422975
.

Files

Files Size Format View
CzechMathJ_62-2012-2_9.pdf 264.7Kb application/pdf View/Open
Back to standard record
Partner of
EuDML logo