Certain differential identities on prime rings and Banach algebras.

Let R be a prime ring and L a Lie ideal of R. The purpose of this paper is to classify generalized derivations F 1 , F 2 , F 3 of R satisfying the following differential identity F 1 (x) ⊥ F 2 (y) = F 3 (x) ⊥ y f o r a l l x , y ∈ L , where ⊥ represents either the Lie product [.,.], or the Jordan product ∘ . Furthermore, as an application, the same identities are studied locally on nonvoid open subsets of prime Banach algebras. [ABSTRACT FROM AUTHOR]