Geodesic
Annihilating and power-commuting generalized skew derivations on Lie ideals in prime rings
Czechoslovak Mathematical Journal, Tome 66 (2016) no. 2, pp. 481-492
Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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 $x\in R$.
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 $x\in R$.
DOI : 10.1007/s10587-016-0270-1
Classification : 16N60, 16W25
Keywords: generalized skew derivation; Lie ideal; prime ring 
@article{10_1007_s10587_016_0270_1,
     author = {de Filippis, Vincenzo},
     title = {Annihilating and power-commuting generalized skew derivations on {Lie} ideals in prime rings},
     journal = {Czechoslovak Mathematical Journal},
     pages = {481--492},
     year = {2016},
     volume = {66},
     number = {2},
     doi = {10.1007/s10587-016-0270-1},
     mrnumber = {3519616},
     zbl = {06604481},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1007/s10587-016-0270-1/}
}
TY  - JOUR
AU  - de Filippis, Vincenzo
TI  - Annihilating and power-commuting generalized skew derivations on Lie ideals in prime rings
JO  - Czechoslovak Mathematical Journal
PY  - 2016
SP  - 481
EP  - 492
VL  - 66
IS  - 2
UR  - http://geodesic.mathdoc.fr/articles/10.1007/s10587-016-0270-1/
DO  - 10.1007/s10587-016-0270-1
LA  - en
ID  - 10_1007_s10587_016_0270_1
ER  - 
%0 Journal Article
%A de Filippis, Vincenzo
%T Annihilating and power-commuting generalized skew derivations on Lie ideals in prime rings
%J Czechoslovak Mathematical Journal
%D 2016
%P 481-492
%V 66
%N 2
%U http://geodesic.mathdoc.fr/articles/10.1007/s10587-016-0270-1/
%R 10.1007/s10587-016-0270-1
%G en
%F 10_1007_s10587_016_0270_1
de Filippis, Vincenzo. Annihilating and power-commuting generalized skew derivations on Lie ideals in prime rings. Czechoslovak Mathematical Journal, Tome 66 (2016) no. 2, pp. 481-492. doi: 10.1007/s10587-016-0270-1

[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). | MR | Zbl

[2] Carini, L., Filippis, V. De: Commutators with power central values on a Lie ideal. Pac. J. Math. 193 (2000), 269-278. | DOI | MR | Zbl

[3] Carini, L., Filippis, V. De, Scudo, G.: Power-commuting generalized skew derivations in prime rings. Mediterr. J. Math. 13 (2016), 53-64. | DOI | MR | Zbl

[4] Chang, J.-C.: On the identity {$h(x)=af(x)+g(x)b$}. Taiwanese J. Math. 7 (2003), 103-113. | DOI | MR | Zbl

[5] Chuang, C. L.: Differential identities with automorphisms and antiautomorphisms. II. J. Algebra 160 (1993), 130-171. | DOI | MR

[6] Chuang, C.-L.: Differential identities with automorphisms and antiautomorphisms. I. J. Algebra 149 (1992), 371-404. | DOI | MR

[7] Chuang, C.-L.: GPIs having coefficients in Utumi quotient rings. Proc. Am. Math. Soc. 103 (1988), 723-728. | DOI | MR | Zbl

[8] Chuang, C.-L., Lee, T.-K.: Identities with a single skew derivation. J. Algebra 288 (2005), 59-77. | DOI | MR | Zbl

[9] Filippis, V. De: Generalized derivations and commutators with nilpotent values on Lie ideals. Tamsui Oxf. J. Math. Sci. 22 (2006), 167-175. | MR | Zbl

[10] Filippis, V. De, Vincenzo, O. M. Di: Vanishing derivations and centralizers of generalized derivations on multilinear polynomials. Commun. Algebra 40 (2012), 1918-1932. | DOI | MR | Zbl

[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. | DOI | MR | Zbl

[12] Dhara, B., Kar, S., Mondal, S.: Generalized derivations on Lie ideals in prime rings. Czech. Math. J. 65 (140) (2015), 179-190. | DOI | MR | Zbl

[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. | MR | Zbl

[14] Herstein, I. N.: Topics in Ring Theory. Chicago Lectures in Mathematics The University of Chicago Press, Chicago (1969). | MR | Zbl

[15] Jacobson, N.: Structure of Rings. Colloquium Publications 37. Amer. Math. Soc. Providence (1964). | MR

[16] Lanski, C.: An Engel condition with derivation. Proc. Am. Math. Soc. 118 (1993), 731-734. | DOI | MR | Zbl

[17] Lanski, C.: Differential identities, Lie ideals, and Posner's theorems. Pac. J. Math. 134 (1988), 275-297. | DOI | MR | Zbl

[18] Lanski, C., Montgomery, S.: Lie structure of prime rings of characteristic 2. Pac. J. Math. 42 (1972), 117-136. | DOI | MR

[19] Lee, T.-K.: Generalized skew derivations characterized by acting on zero products. Pac. J. Math. 216 (2004), 293-301. | DOI | MR | Zbl

[20] Lee, T.-K., Liu, K.-S.: Generalized skew derivations with algebraic values of bounded degree. Houston J. Math. 39 (2013), 733-740. | MR | Zbl

[21] III., W. S. Martindale: Prime rings satisfying a generalized polynomial identity. J. Algebra 12 (1969), 576-584. | DOI | MR | Zbl

[22] Posner, E. C.: Derivations in prime rings. Proc. Am. Math. Soc. 8 (1957), 1093-1100. | DOI | MR

[23] Rowen, L. H.: Polynomial Identities in Ring Theory. Pure and Applied Math. 84 Academic Press, New York (1980). | MR | Zbl

[24] Wang, Y.: Power-centralizing automorphisms of Lie ideals in prime rings. Commun. Algebra 34 (2006), 609-615. | DOI | MR | Zbl

[25] Wong, T.-L.: Derivations with power-central values on multilinear polynomials. Algebra Colloq. 3 (1996), 369-378. | MR | Zbl

Cité par Sources :