Commuting and Centralizing Generalized Derivations on Lie Ideals in Prime Rings

Let $R$ be a noncommutative prime ring of characteristic different from $2$, $U$ the Utumi quotient ring of $R$, $C$ the extended centroid of $R$, and $L$ a noncentral Lie ideal of $R$. If $F$ and $G$ are generalized derivations of $R$ and $k\ge1$ a fixed integer such that $[F(x),x]_kx-x[G(x),x]_k=0$ for any $x\in L$, then one of the following holds:

• 1) either there exists an $a\in U$ and an $\alpha\in C$ such that $F(x)=xa$ and $G(x)=(a+\alpha)x$ for all $x\in R$;
• 2) or $R$ satisfies the standard identity $s_4(x_1,\dots,x_4)$ and one of the following conclusions occurs: \begin{itemize}
• (a) there exist $a,b,c,q\in U$, such that $a-b+c-q\in C$ and $F(x)=ax+xb$, $G(x)=cx+xq$ for all $x\in R$;
• (b) there exist $a,b,c\in U$ and a derivation $d$ of $U$ such that $F(x)=ax+d(x)$ and $G(x)=bx+xc-d(x)$ for all $x\in R$, with $a+b-c\in C$.

\end{itemize}