Generalized Derivations Commuting on Lie Ideals in Prime Rings

Abstract

Let R be a prime ring of characteristic different from 2, U its Utumi quotient ring, C its extended centroid, L a non-central Lie ideal of R and F, G and H three generalized derivations of R. If

$$\begin{aligned}{}[F(u)G(u)-uH(u),u]=0 \end{aligned}$$

for all $u \in L$, then one of the following holds:

• (1) there exist $a,c,q,p,p'\in U$ such that $F(x)=ax$, $G(x)=cx+xq$ and $H(x)=px+xp'$, for any $x\in R$, with $a, ac-p, p'-aq\in C$;

• (2) there exist $a,c,p\in U$ such that $F(x)=xa$, $G(x)=cx$ and $H(x)=px$, for any $x\in R$, with $ac-p\in C$;

• (3) there exist $c,p\in U$, $\lambda ,\mu \in C$ and a derivation h of R such that $F(x)=\mu x$, $G(x)=cx+\lambda h(x)$ and $H(x)=px+h(x)$, for any $x\in R$, with $\mu c-p\in C$ and $\lambda \mu =1$;

• (4) R satisfies $s_4$, the standard identity of degree 4.