Let 𝑅 be a prime ring with its Utumi ring of quotient $U,H$ and 𝐺 be two generalized derivations of 𝑅 and 𝐿 a noncentral Lie ideal of 𝑅. Suppose that there exists $0\neq a\in R$ such that $a(H(u)u-uG(u))^n=0$ for all $u\in L$, where $n\geq 1$ is a fixed integer. Then there exist $b',c' \in U$ such that $H(x)=b'x+xc',G(x)=c'x$ for all $x\in R$ with $ab'=0$, unless 𝑅 satisfies $s4$, the standard identity in four variables.

Let $R$ be a prime ring of characteristic $\neq 2$, $Qr$ its right Martindale quotient ring, $C$ its extended centroid, $F \neq 0$ a generalized skew derivation of $R, f (x_{1}, . . . , x_{n})$ a multilinear polynomial over $C$ not central-valued on $R$ and $S$ the set of all evaluations of $f (x_{1}, . . . , x_{n})$ in $R$. If $a[F(x), x] \in C$ for all $x \in S$, then there exist $\lambda \in C$ and $b \in Qr$ such that $F(x) = bx + xb + \lambda x$, for all $x \in R$ and one of the following holds:(1) $b \in C$;(2) $f (x_{1}, . . . , x_{n})^{2}$ is central-valued on $R$;(3) $R$ satisfies $s_{4}$, the standered identity of degree 4.