Generalized derivations as homomorphisms or anti-homomorphisms on Lie ideals. (Q903346)

Let \(R\) be a prime ring with \(\text{char\,}R\neq 2\), noncentral Lie ideal \(L\), and derivation \(d\). The main result of this paper is that if \(F\colon R\to R\) is an additive map so that for all \(x,y \in R\), \(F(xy)=F(x)y+xd(y)\), then \(F=0\) if \(F(st)=F(s)F(t)\) for all \(s,t\in L\), or instead if \(F(st)=F(t)F(s)\).