On Lie Structure of Prime Rings with Generalized (alpha, beta)-Derivations

Let R be a ring and alpha, beta be automorphisms of R. An additive mapping F: R -> R is called a generalized (alpha, beta)-derivation on R if there exists an (alpha, beta) derivation d: R -> R such that F(xy) = F(x)alpha(y) beta(x)d(y) holds for all x, y is an element of R. For any x, y is an element of R, set [x, y](alpha, beta) = x alpha(y) beta(y)x and (x o y)(alpha,beta) = x alpha(y) beta(y)(x). In the present paper, we shall discuss the commutativity of a prime ring R admitting generalized (alpha, beta) -derivations F and G satisfying any one of the following properties: (i) F([x, yl) = (x o, (ii) F(x o y)(alpha, beta) = [x, y](alpha, beta) (iii) [F(x), y](alpha, beta) = (F(x) o y)(alpha, beta), (iv) F([x, yl) = [F(x), y](alpha, beta) (v) F(x o y) = (F(x) oy)(alpha, beta), (vi) F([x, y] = [alpha(x), G(y)] and (vii) F(x o y) = (alpha(x) o G(y)) for all x, y in some appropriate subset of R. Finally, obtain some results on semi-projective Morita context with generalized (alpha, beta)-derivations.