Skew-commuting and commuting mappings in rings

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 × R → 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 re-skew-commuting on H, then g(H) = {0}. The second main part: Let re ≥ 2. If R is an n!-torsion-free semiprime ring, and d : R → R is a derivation such that d2 is re-commuting on R, then d maps R into its center. © Birkhäuser Verlag, 2002.