An Engel condition with an additive mapping in semiprime rings. (Q2018812)

The aim of this article is to prove the following result: Let \(n>1\) be a fixed integer, let \(R\) be an \(n!\)-torsion free semiprime ring, and let \(f\colon R\to R\) be an additive mapping satisfying the relation \([f(x),x]_n=[[\cdots[[f(x),x],x],\cdots],x]=0\) for all \(x\in R\). Then \(f\) is a commuting map of \(R\), that is, \([f(x),x]=0\) for all \(x\in R\). Since von Neumann algebras and \(C^*\) algebras are always semiprime, this purely algebraic result might be of some interest from functional analysis point of view.