Structure of weakly periodic rings with potent extended commutators (Q5931276)

Let \(n>1\) be a fixed positive integer. Let \(R\) denote a ring, \(N\) its set of nilpotent elements, and \(P=\{x\in R\mid x^m=x\) for some \(m>1\}\). Call \(R\) weakly periodic if \(R=P+N\). Suppose that \(R\) is weakly periodic and for each \(x_1,x_2,\dots,x_n\in R\setminus N\) there exists \(\sigma\in S_n\) such that \(\sigma(n)\neq n\), and a positive integer \(k\) such that the extended commutator \([x_1x_2\cdots x_n,x_{\sigma(1)}x_{\sigma(2)}\cdots x_{\sigma(x)}]_k\in P\). It is proved that \(R\) is periodic and the commutator ideal \(C(R)\) is nil. Moreover, under the additional hypotheses that all \(\sigma\) satisfy \(\sigma(1)=n\) and \(\sigma(n)=1\) and \([a,b]\in P\) for all \(a,b\in N\), it is proved that \(R\) must be commutative.