On Lie Ideals with Generalized Derivations and Non-commutative Banach Algebras

Let R be a prime ring of characteristic different from 2, L be a non-central Lie ideal of R and m, n be the fixed positive integers. If R admits a generalized derivation F associated with a deviation d such that F(u(2))(m) - d(u)(2n) is an element of Z( R) for all u is an element of L, then R satisfies s4, the standard identity in four variables. Moreover, we also examine the case when R is semiprime ring. Finally, as an application we obtain some range inclusion results of continuous or spectrally bounded generalized derivations on Banach algebras.