RINGSWITH (R,R,R)AND [R,(R,R,R)] IN THE LEFT NUCLEUS

Let $R$ be a nonassociative ring, and $N = (R,R,R) + [R,(R,R,R)]$. We show that $W = \{w\in N | Rw +wR +R(wR) \subset N\}$ is a two-sided ideal of $R$. If for some $r\in R$, any one of the sets $(r, R, R)$, $(R,r, R)$ or $(R, R,r)$ is contained in $W$, then the other two sets are contained in $W$ also. If the associators are assumed to be contained in either the left, the middle, or the right nucleus, and $I$ is the ideal generated by all associators, then $I^2 \subset W$. If $N$ is assumed to be contained in the left or the right nucleus, then $W^2 = 0$. We conclude that if $R$ is semiprime and $N$ is contained in the left or the right nucleus, then $R$ is associative. We assume characteristic not 2.