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

Czechoslovak Mathematical Journal

Vincenzo De Filippis¹ 

[1] Department of Mathematics and Computer Science, University of Messina, Viale Stagno d'Alcontres 31, 98166 Messina, Italy

关键词: generalized skew derivation;    Lie ideal;    prime ring;   

DOI  :  

学科分类：数学（综合）

来源: Akademie Ved Ceske Republiky

【 摘 要 】

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 $n\geq1$ a fixed positive integer. Let $\alpha$ be an automorphism of the ring $R$. An additive map $D R\to\nobreak R$ 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 $F R\to R$ 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 $x\in R$.

【 授权许可】

Unknown