RINGS WITH A DERIVATION WHOSE IMAGE IS CONTAINED IN THE NUCLEI
Keywords:Nonassociative ring, nucleus, derivation, d-invariant, semiprime ring, prime ring, simple ring
Let $R$ be a nonassociative ring, $N$, $M$, $L$ and $G$ the left nucleus, middle nucleus, right nucleus and nucleus respectively. Suh  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  and Yen  for the prime nonassociative rings. In our recent papers we partially generalize the above main result.
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