Previous |  Up |  Next

Article

Title: Annihilating and power-commuting generalized skew derivations on Lie ideals in prime rings (English)
Author: de Filippis, Vincenzo
Language: English
Journal: Czechoslovak Mathematical Journal
ISSN: 0011-4642 (print)
ISSN: 1572-9141 (online)
Volume: 66
Issue: 2
Year: 2016
Pages: 481-492
Summary lang: English
.
Category: math
.
Summary: Let $R$ be a prime ring of characteristic different from 2 and 3, $Q_r$ its right Martindale quotient ring, $C$ its extended centroid, $L$ a non-central Lie ideal of $R$ and $n\geq 1$ a fixed positive integer. Let $\alpha $ be an automorphism of the ring $R$. An additive map $D\colon R\to R$ is called an $\alpha $-derivation (or a skew derivation) on $R$ if $D(xy)=D(x)y+\alpha (x)D(y)$ for all $x,y\in R$. An additive mapping $F\colon R\to R$ is called a generalized $\alpha $-derivation (or a generalized skew derivation) on $R$ if there exists a skew derivation $D$ on $R$ such that $F(xy)=F(x)y+\alpha (x)D(y)$ for all $x,y\in R$. We prove that, if $F$ is a nonzero generalized skew derivation of $R$ such that $F(x)\* [F(x),x]^n = 0$ for any $x\in L$, then either there exists $\lambda \in C$ such that $F(x)=\lambda x$ for all $x\in R$, or $R\subseteq M_2(C)$ and there exist $a\in Q_r$ and $\lambda \in C$ such that $F(x)=ax+xa+\lambda x$ for any $x\in R$. (English)
Keyword: generalized skew derivation
Keyword: Lie ideal
Keyword: prime ring
MSC: 16N60
MSC: 16W25
.
Date available: 2016-06-16T12:57:41Z
Last updated: 2016-06-20
Stable URL: http://hdl.handle.net/10338.dmlcz/145738
.
Reference: [1] Beidar, K. I., III., W. S. Martindale, Mikhalev, A. V.: Rings with Generalized Identities.Pure and Applied Mathematics 196 Marcel Dekker, New York (1996). Zbl 0847.16001, MR 1368853
Reference: [2] Carini, L., Filippis, V. De: Commutators with power central values on a Lie ideal.Pac. J. Math. 193 (2000), 269-278. Zbl 1009.16034, MR 1755818, 10.2140/pjm.2000.193.269
Reference: [3] Carini, L., Filippis, V. De, Scudo, G.: Power-commuting generalized skew derivations in prime rings.Mediterr. J. Math. 13 (2016), 53-64. MR 3456907, 10.1007/s00009-014-0493-z
Reference: [4] Chang, J.-C.: On the identity {$h(x)=af(x)+g(x)b$}.Taiwanese J. Math. 7 (2003), 103-113. Zbl 1048.16018, MR 1961042
Reference: [5] Chuang, C. L.: Differential identities with automorphisms and antiautomorphisms. II.J. Algebra 160 (1993), 130-171. MR 1237081, 10.1006/jabr.1993.1181
Reference: [6] Chuang, C.-L.: Differential identities with automorphisms and antiautomorphisms. I.J. Algebra 149 (1992), 371-404. MR 1172436, 10.1016/0021-8693(92)90023-F
Reference: [7] 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: [8] Chuang, C.-L., Lee, T.-K.: Identities with a single skew derivation.J. Algebra 288 (2005), 59-77. Zbl 1073.16021, MR 2138371, 10.1016/j.jalgebra.2003.12.032
Reference: [9] Filippis, V. De: Generalized derivations and commutators with nilpotent values on Lie ideals.Tamsui Oxf. J. Math. Sci. 22 (2006), 167-175. Zbl 1133.16022, MR 2285443
Reference: [10] Filippis, V. De, Vincenzo, O. M. Di: Vanishing derivations and centralizers of generalized derivations on multilinear polynomials.Commun. Algebra 40 (2012), 1918-1932. Zbl 1258.16043, MR 2945689, 10.1080/00927872.2011.553859
Reference: [11] Filippis, V. De, Scudo, G.: Strong commutativity and Engel condition preserving maps in prime and semiprime rings.Linear Multilinear Algebra 61 (2013), 917-938. Zbl 1281.16045, MR 3175336, 10.1080/03081087.2012.716433
Reference: [12] Dhara, B., Kar, S., Mondal, S.: Generalized derivations on Lie ideals in prime rings.Czech. Math. J. 65 (140) (2015), 179-190. MR 3336032, 10.1007/s10587-015-0167-4
Reference: [13] Vincenzo, O. M. Di: On the {$n$}-th centralizer of a Lie ideal.Boll. Unione Mat. Ital., A Ser. (7) 3 (1989), 77-85. Zbl 0692.16022, MR 0990089
Reference: [14] Herstein, I. N.: Topics in Ring Theory.Chicago Lectures in Mathematics The University of Chicago Press, Chicago (1969). Zbl 0232.16001, MR 0271135
Reference: [15] Jacobson, N.: Structure of Rings.Colloquium Publications 37. Amer. Math. Soc. Providence (1964). MR 0222106
Reference: [16] Lanski, C.: An Engel condition with derivation.Proc. Am. Math. Soc. 118 (1993), 731-734. Zbl 0821.16037, MR 1132851, 10.1090/S0002-9939-1993-1132851-9
Reference: [17] 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: [18] Lanski, C., Montgomery, S.: Lie structure of prime rings of characteristic 2.Pac. J. Math. 42 (1972), 117-136. MR 0323839, 10.2140/pjm.1972.42.117
Reference: [19] Lee, T.-K.: Generalized skew derivations characterized by acting on zero products.Pac. J. Math. 216 (2004), 293-301. Zbl 1078.16038, MR 2094547, 10.2140/pjm.2004.216.293
Reference: [20] Lee, T.-K., Liu, K.-S.: Generalized skew derivations with algebraic values of bounded degree.Houston J. Math. 39 (2013), 733-740. Zbl 1285.16038, MR 3126322
Reference: [21] III., W. S. Martindale: Prime rings satisfying a generalized polynomial identity.J. Algebra 12 (1969), 576-584. Zbl 0175.03102, MR 0238897, 10.1016/0021-8693(69)90029-5
Reference: [22] Posner, E. C.: Derivations in prime rings.Proc. Am. Math. Soc. 8 (1957), 1093-1100. MR 0095863, 10.1090/S0002-9939-1957-0095863-0
Reference: [23] Rowen, L. H.: Polynomial Identities in Ring Theory.Pure and Applied Math. 84 Academic Press, New York (1980). Zbl 0461.16001, MR 0576061
Reference: [24] Wang, Y.: Power-centralizing automorphisms of Lie ideals in prime rings.Commun. Algebra 34 (2006), 609-615. Zbl 1093.16020, MR 2211941, 10.1080/00927870500387812
Reference: [25] Wong, T.-L.: Derivations with power-central values on multilinear polynomials.Algebra Colloq. 3 (1996), 369-378. Zbl 0864.16031, MR 1422975
.

Fulltext not available (moving wall 24 months)

Partner of
EuDML logo