RINGS WITH ASSOCIATORS IN THE LEFT AND MIDDLE NUCLEUS

Let $R$ be a nonassociative ring, $N$, $M$ and $L$ the left, middle and right nucleus respectively. It is shown that if $R$ a semipnme ring satisfying $(R,R,R) \subset N\cap M$ (resp. $(R,R,R) \subset M\cap L$), then $L\subset M\subset N$(resp. $N\subset M\subset L$); moreover, $R$ is associative if $((R,R,M),(R,R,R)) = 0$ (resp. $((M,R,R),(R,R,R)) = 0)$ or $(M,R) \subset M$; and the Abelian group $(R,+ )$ has no elements of order 2. We also prove that if $R$ is a simple ring satisfying char $R \neq 2$, and $(R, R, R) \subset N \cap M$ or $(R, R, R) \subset M \cap L$ then $R$ is associative.