A note on rings with certain variable identities (Q1123964)

For the ring R, define extended commutators \([x_ 1,x_ 2,...,x_ k]\) by taking \([x_ 1,x_ 2]=x_ 1x_ 2-x_ 2x_ 1\) and letting \([x_ 1,...,x_ k]=[[x_ 1,...,x_{k-1}],x_ k]\) for \(k>2\). If \(x_ 1=x\) and \(x_ 2=...=x_ k=y\), write \([x,y]_ k\) instead of [x,y,...,y]. \textit{A. A. Klein, \textit{I. Nada}}, and the reviewer [Bull. Aust. Math. Soc. 22, 285-289 (1980; Zbl 0442.16030)] conjectured that if \(k>1\) is a fixed integer and for each \(x,y\in R\) there exist positive integers m, n such that \([x^ m,y^ n]_ k=0\), then the commutator ideal of R must be nil; and they proved the conjecture for R with 1. The case \(k=2\) for arbitrary rings had been proved earlier by \textit{I. N. Herstein} [J. Algebra 38, 112-118 (1976; Zbl 0323.16014)]. In the present note, the conjecture is proved for Artinian rings and periodic rings; and it is shown that the same hypotheses imply commutativity in prime rings having nonzero idempotents or nonzero center. Also included are two nil- commutator-ideal theorems for rings in which certain extended commutators are central.