Prime Lie Rings of Generalized Derivations of Commutative Rings

Let R be a commutative ring with identity. By a Bres̃ar generalized derivation of R we mean an additive map g : R→ R such that g (xy) = g (x) y + xd (y) for all x, y ∈ R, where d is a derivation of R. And an additive mapping f : R → R is called a generalized derivation in the sense of Nakajima if it satisfies f(xy) = f(x)y + xf(y) − xf(1)y for all x, y ∈ R. In this paper we extend some results of Chebotar and Lee [2] and, Liu and Passman [5] to generalized derivations of Nakajima and Bres̃ar. The main aim of this note is to give some properties of the Lie ring Rg which is the set of all Bres̃ar generalized derivations of R of the form rg with r ∈ R and is to apply similar results to generalized derivations of Nakajima.