PRIME NONASSOCIATIVE RINGS WITH SKEW DERIVATIONS

Let $R$ be a prime nonassociative ring, $G$ the nucleus of $R$ and $s$, $t$ be automorphisms of $R$.(I) Suppose that $\delta$ is an $s$-derivation of $R$ such that $s\delta=\delta s$ and $\lambda$ is an $t$-derivation of $R$. If $\lambda\delta^n=0$ and $\delta^n(R)\subset G$, where $n$ is a fixed positive integer, then $\lambda=0$ or $\delta^{3n-1}=0$.(II) Assume that $\delta$ and $\lambda$ are derivations of $R$. If there exists a fixed positive integer $n$ such that $\lambda^n\delta=0$, and $\delta(R)\subset G$ or $\lambda^n(R)\subset G$, then $\delta^2= 0$ or $\lambda^{6n-4}= 0$.