Skew-commuting and commuting mappings in rings

Abstract

We study some maps which are skew-commuting or skew-centralizing on additive subgroups of rings with a left identity; and we present some results concerning commuting mappings in semiprime rings.¶ The first main part: Let n denote an arbitrary positive integer. Let R be a ring with left identity e, and let H be an additive subgroup of R containing e. Let \( G : R \times R \to R \) be a symmetric bi-additive mapping and let \( g \) be the trace of G. Let R be n!-torsion-free if \( n > 1 \), and 2-torsion-free if n = 1. If \( g \) is n-skew-commuting on H, then \( g(H) = \{0 \} \).¶ The second main part: Let \( n \geq 2 \). If R is an n!-torsion-free semiprime ring, and \( d: R \to R \) is a derivation such that d2 is n-commuting on R, then d maps R into its center.