Rings with generalized commutators in the nuclei

Abstract

Let $ R$ be a prime weakly Novikov ring and $ T_k=\underbrace{[[[\ldots[[R,R],R]\ldots,R],R],R]}_{k R's}$ where $ k$ is a positive integer. We prove that if $ T_k\subseteq N_l\cap N_r$ or $ T_k\subseteq N_m\cap N_r$ then $ R$ is associative or $ T_k=0$. Moreover, if $ T_k$ is contained in two of the three nuclei, and $ k=2$ or $ k=3$ then the same conclusions hold. We also consider such rings with derivations. Some similar results of weakly M-rings are obtained.