RINGS WITH $(x,R,x)$ AND $(N +NR,R)$ IN THE LEFT NUCLEUS
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Let $R$ be a nonassociative ring, $N$ the left nucleus. Assume thatN is a nonzero Lie ideal of $R$. It is shown that if $R$ is a prime ring which satisfies $(x,R,x) \subset N$ and $(NR,R)\subset N$ then $R$ is either associative or commutative.
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