On skew-commuting mappings of rings

Abstract

A mapping f of a ring R into itself is called skew-commuting on a subset S of R if f(s)s + sf(s) = 0 for all s ∈ S. We prove two theorems which show that under rather mild assumptions a nonzero additive mapping cannot have this property. The first theorem asserts that if R is a prime ring of characteristic not 2, and f: R → R is an additive mapping which is skew-commuting on an ideal I of R, then f(I) = 0. The second theorem states that zero is the only additive mapping which is skew-commuting on a 2-torsion free semiprime ring.