Strong commutativity preserving maps on Lie ideals

Abstract

Let A be a prime ring and let R be a noncentral Lie ideal of A . An additive map f : R → A is called strong commutativity preserving (SCP) on R if [ f ( x ) , f ( y ) ] = [ x , y ] for all x , y ∈ R . In this paper we show that if f is SCP on R , then there exist λ ∈ C , λ 2 = 1 and an additive map μ : R → Z ( A ) such that f ( x ) = λ x + μ ( x ) for all x ∈ R where C is the extended centroid of A , unless char A = 2 and A satisfies the standard identity of degree 4.