NONASSOCIATIVE RINGS WITH A SPECIAL DERIVATION

Let $R$ be a nonassociative ring, $N$, $L$ and $G$ the left nucleus, right nucleus and nucleus respectively. It is shown that if $R$ is a prime ring with a derivafion $d$ such that $ax+ d(x) \in G$ where $a \in Z$, the ring of rational integers, or $a \in G$ with $(ad)(x) = ad(x) = d(ax)$ and $ax = xa$ for all $x$ in $R$ then either $R$ is associative or $ad+ d^2 = 2d(R)^2 = 0$. This result is also valid under the weaker hypothesis $ax+ d(x) \in N \cap L$ for all $x$ in $R$ for the simple ring case, and we obtain that either $R$ is associative or $((ad+ d^2)(R))^2 =0$ for the prime ring case.