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Generalized Skew Derivations With Engel Conditions On Lie Ideals
by Jui-Chi Chang

Vol. 6 No. 3 (2011) P.305~P.320

ABSTRACT

Let $R$ be a prime ring and $L$ a noncommutative Lie ideal of $R$. Suppose that $f$ is a nonzero right generalized $\beta$-derivation of $R$ associated with a $\beta$-derivation $\delta$ such that $[f(x),x]_k=0$ for all $x\in L$, where $k$ is a fixed positive integer. Then either there exists $s\in C$ scuh that $f(x)=sx$ for all $x\in R$ or $R\subseteq M_2(F)$ for some field $F$. Moreover, if the latter case holds, then either ${\rm char} R= 2$ or ${\rm char}R\ne 2$ and $f(x)=bx-xc$ for all $x\in R$, where $b,c\in\mathop{_{\mathscr F}\hspace{-0.3mm}R}$ and $b+c\in C$.

KEYWORDS
Skew derivation, generalized skew derivation, automorphism, prime ring, generalized polynomial identity (GPI), Lie ideal, Engel condition

MATHEMATICAL SUBJECT CLASSIFICATION 2010
Primary: 16W20, 16W25, 16W55

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