An Engel condition with an additive mapping in semiprime rings

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 → R be an additive mapping satisfying the relation ( f( x), x)n =( ( ... (( f( x), x) ,x ) , . .. ) ,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.