Abstract
Let R be a 2-torsion free prime ring with center Z, U be the Utumi quotient ring, Q be the Martindale quotient ring of R, d be a derivation of R and L be a Lie ideal of R. If d(uv)(n) - d(u)(m)d(v)(l) or d(uv)(n) = d(v)(l)d(u)(m) for all u, v is an element of L, where m, n, l are fixed positive integers, then L subset of Z. We also examine the case when R is a semiprime ring. Finally, as an application we apply our result to the continuous derivations on non commutative Banach algebras. This result simultaneously generalizes a number of results in the literature.