RINGS WITH A DERIVATION WHOSE IMAGE IS CONTAINED IN THE NUCLEI

Let $R$ be a nonassociative ring, $N$, $M$, $L$ and $G$ the left nucleus, middle nucleus, right nucleus and nucleus respectively. Suh [4] proved that if $R$ is a prime ring with a derivation dsuch that $d(R) \subseteq G$ then either $R$ is associative or $d^3 =0$. We improve this result by concluding that either $R$ is associative or $d^2 =2d =0$ under the weaker hypothesis $d(R)\subseteq N$\cap M$ or $d(R)\subseteq N\cap M$ or $d(R)\subseteq M\cap L$. Using our result, we obtain the theorems of Posner [3] and Yen [11] for the prime nonassociative rings. In our recent papers we partially generalize the above main result.