RINGS WITH A JORDAN DERIVATION WHOSE IMAGE IS CONTAINED IN THE NUCLEI OR COMMUTATIVE CENTER

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 Jordan derivation d such that $d(R) \subseteq G$ and $(d^2 R), R) \subseteq N$ or $(d^2(R), R) \subseteq L$ then either $R$ is associative or $2d^2 =0$. Moreover. if $(d(R), R) =0$ then either $R$ is associative and commutative or $2d =0$. We also prove that if $R$ is a prime ring with a derivation $d$ and there exists a fixed positive integer $n$ such that $d^n(R) \subseteq G$ and $(d^n(R), R) =0$ then $R$ is associative and $d^n =0$, or $R$ is associative and commutative, or $d^{2n} = (\frac{(2n)!}{n!})d^n = 0$. This partially generalize the results of [3]. We also obtain some results on prime rings with a derivation satisfying other hypotheses.