An Engel condition with an additive mapping in semiprime rings

Abstract

The main purpose of this paper is to prove the following result: Let n > 1 be a fixed integer, let R be a n!-torsion free semiprime ring, and let \(f:R\rightarrow R\) be an additive mapping satisfying the relation \([f(x),x]_{n}= [[{\ldots } [[ f(x),x] ,x] ,{\ldots } ],x]=0\) for all x ∈ R. In this case [f(x), x] = 0 is fulfilled for all x ∈ R. Since any semisimple Banach algebra (for example, C ∗ algebra) is semiprime, this purely algebraic result might be of some interest from functional analysis point of view.