On Lie derivations of Lie ideals of prime algebras

Let F be a commutative ring with 1, let A be a prime F-algebra with Martindale extended centroid C and with central closure A(c) and let R be a noncentral Lie ideal of the algebra A generating A. Further, let Z(R) be the center of R, let (R) over bar = R/Z(R) be the factor Lie algebra and let delta: (R) over bar --> (R) over bar be a Lie derivation. Suppose that char(A) not equal 2 and A does not satisfy St(14), the standard identity of degree 14. We show that R boolean AND C = Z(R) and there exists a derivation of algebras D: A --> A(c) such that x(D) + C = (x + C)(delta) is an element of (R + C)/C = (R) over bar for all x is an element of R. Our result solves an old problem of Herstein.