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# On Lie Structure of Prime Rings with Generalized (α, β)-Derivations

Author(s): Nadeem ur Rehman | Radwan Mohammed | AL-Omary | Claus Haetinger

Journal: Boletim da Sociedade Paranaense de Matemática
ISSN 0037-8712

Volume: 27;
Issue: 2;
Start page: 43;
Date: 2009;
Original page

Keywords: Lie ideals | prime rings | | β)-derivations | generalized (α | β)-derivations | Morita context | reduced rings.

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.