RINGS WITH A DERIVATION WHOSE IMAGE IS ZERO ON THE ASSOCIATORS

Abstract


 
 
 Let $R$ be a nonassociative ring, $N$, $M$, $L$ and $G$ the left nucleus,middle nucleus, right nucleus and nucleus respectively. Assume that $R$ is a ring with a derivation $d$ such that $d((R, R, R)) = 0$. It is shown that if $R$ is a simple ring then either $R$ is associative or $d(N \cap L) = 0$; and if $R$ is a prime ring satisfying $Rd(G) \subseteq N$ and $d(G)R \subseteq L$, or $d(G)R +Rd(G) \subseteq M$ then either $R$ is associative or $d(G) =0$. These partially extend our previous results.