Annihilator of Power Values of Generalized Derivations in Prime Rings

Let R be a prime ring of characteristic different from 2, U its Utumi quotient ring, C the center of U, F a non-zero generalized derivation of R and L a non-commutative Lie ideal of R. Suppose that there exists $$0\ne a\in R$$ such that $$a(u^s[F(u),u]u^t)^n=0$$ for all $$u \in L$$ , where $$s\ge 0, t\ge 0, n\ge 1$$ are fixed integers. Then either $$F(x)=\alpha x$$ for all $$x\in R$$ with $$\alpha \in C$$ or R satisfies $$s_4(x_1,\ldots ,x_4)$$ , the standard identity in four variables, and $$F(x)=bx+xb+\alpha x$$ for all $$x\in R$$ , for some $$b\in U$$ and $$\alpha \in C$$ .