| 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 |
| idZBL:
|
Zbl 06604481 |
| idMR:
|
MR3519616 |
| DOI:
|
10.1007/s10587-016-0270-1 |
| . |
| Date available:
|
2016-06-16T12:57:41Z |
| Last updated:
|
2020-07-03 |
| 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. Zbl 1342.16039, 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, 10.11650/twjm/1500407520 |
| 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. Zbl 1337.16033, 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 |
| . |
