Boletim da Sociedade Paranaense de Matemática (Nov 2009)
On Lie Structure of Prime Rings with Generalized (α, β)-Derivations
Abstract
Let R be a ring and α, β be automorphisms of R. An additive mappingF: R → R is called a generalized (α,β)-derivation on R if there exists an (α,β)-derivation d:R → R such that F(xy)=F(x)α(y) + β(x(y) holds for all x, y ∈ R. For any x, y ∈ R, set [x, y]_{α,β} = xα(y)−(y)x and (x◦y)_{α,β} = xα(y)+β(y)x. In the present paper, we shall discuss the commutativity of a prime ring R admitting generalized (α,β)-derivations F and G satisfying any one of the following properties:(i) F([x, y]) = (x◦y)_{α,β}, (ii) F(x◦y) = [x,y]α,β, (iii) [F(x),y]_{α,β}=(F(x)◦y)_{α,β}, (iv) F([x, y]) = [F(x), y]_{α,β}, (v) F(x◦y) = (F(x◦y)_{α,β}, (vi) F([x,y]=[α(x),G(y)] and (vii) F(x◦y)=(α(x)◦G(y)) for all x, y in some appropriatesubset of R. Finally, obtain some results on semi-projective Morita context with generalized (α, β)-derivations.
