Strong commutativity preserving maps on Lie ideals of semiprime rings. (Q945961)

Let \(R\) be a 2-torsion free semiprime ring with center \(Z\) and nonzero Lie ideal \(U\) so that \(u\in U\) implies that \(u^2\in U\). The authors show that if \(f\colon R\to R\) is a homomorphism or an antihomomorphism so that \(f(U)=U\) then the following are equivalent: 1) \([f(u),u]\in Z\) for all \(u\in U\); and 2) \([f(u),f(v)]=[u,v]\) for all \(u,v\in U\).