A note on generalized Lie derivations of prime rings

Abstract

Let R be a prime ring of characteristic not 2, A be an additive subgroup of R, and F, T, D,K: A → R be additive maps such that F[x, y]) = F(x)y − yK(x) − T(y)x + xD(y) for all x, y ∈ A. Our aim is to deal with this functional identity when A is R itself or a noncentral Lie ideal of R. Eventually, we are able to describe the forms of the mappings F, T, D, and K in case A = R with deg(R)>3 and also in the case A is a noncentral Lie ideal and deg(R)>9. These enable us in return to characterize the forms of both generalized Lie derivations, D-Lie derivations and Lie centralizers of R under some mild assumptions. Finally, we give a generalization of Lie homomorphisms on Lie ideals.

Key words： Prime ring; derivation; generalized derivation; generalized Lie derivation; functional identity; generalized polynomial identity

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