Annihilating and power-commuting generalized skew derivations on Lie ideals in prime rings

Vincenzo de Filippis

Czechoslovak Mathematical Journal (2016)

  • Volume: 66, Issue: 2, page 481-492
  • ISSN: 0011-4642

Abstract

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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 ≥ 1 a fixed positive integer. Let α be an automorphism of the ring R . An additive map D : R → R is called an α -derivation (or a skew derivation) on R if D ( x y ) = D ( x ) y + α ( x ) D ( y ) for all x , y ∈ R . An additive mapping F : R → R is called a generalized α -derivation (or a generalized skew derivation) on R if there exists a skew derivation D on R such that F ( x y ) = F ( x ) y + α ( x ) D ( y ) for all x , y ∈ 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 ∈ L , then either there exists λ ∈ C such that F ( x ) = λ x for all x ∈ R , or R ⊆ M 2 ( C ) and there exist a ∈ Q r and λ ∈ C such that F ( x ) = a x + x a + λ x for any x ∈ R .

How to cite

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de Filippis, Vincenzo. "Annihilating and power-commuting generalized skew derivations on Lie ideals in prime rings." Czechoslovak Mathematical Journal 66.2 (2016): 481-492. <http://eudml.org/doc/280095>.

@article{deFilippis2016,
abstract = {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 $n\ge 1$ a fixed positive integer. Let $\alpha $ be an automorphism of the ring $R$. An additive map $D\colon R\rightarrow R$ 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 $F\colon R\rightarrow R$ 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 $x\in R$.},
author = {de Filippis, Vincenzo},
journal = {Czechoslovak Mathematical Journal},
keywords = {generalized skew derivation; Lie ideal; prime ring},
language = {eng},
number = {2},
pages = {481-492},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Annihilating and power-commuting generalized skew derivations on Lie ideals in prime rings},
url = {http://eudml.org/doc/280095},
volume = {66},
year = {2016},
}

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
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 66
IS - 2
SP - 481
EP - 492
AB - 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 $n\ge 1$ a fixed positive integer. Let $\alpha $ be an automorphism of the ring $R$. An additive map $D\colon R\rightarrow R$ 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 $F\colon R\rightarrow R$ 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 $x\in R$.
LA - eng
KW - generalized skew derivation; Lie ideal; prime ring
UR - http://eudml.org/doc/280095
ER -

References

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