Some identities involving generalized (α, β)-derivations in prime and semiprime rings

Let R be a prime or semiprime ring, lambda a nonzero left-sided ideal of R and alpha, beta be two auto-morphisms of R. Let F : R -> R and G : R -> R he two generalized (alpha, beta)-derivations of R associated with (alpha, beta)-derivations d : R -> R and g : R -> R, respectively. The purpose of this paper is to investigate the following identities: (1) G(xy) + d(x)F(y) + alpha(xy) = 0; (2) G(xy) + d(x)F(y) = 0; (3) G(xy) + d(x)F(y) + alpha(yx) = 0; (4) G(xy) + d(x)F(y) + alpha(yx) + alpha(xy) = 0; (5) G(xy) + d(x)F(y) + alpha(yx) - alpha(xy) = 0; (6) G(xy) + d(y)F(x) + alpha(yx) = 0; (7) G(xy) d(y)F(x) + alpha(yx) + alpha(xy) = 0; (8) G(xy) + d(y)F(x) + alpha(yx) - alpha(xy) = 0; (9) G(xy) + F(x)F(y) = 0; (10) G(xy) + F(y)F(x) = 0; (11) F(xy) + G(x)alpha(y) + alpha(yx) = 0; (12) F(x)F(y) + G(x)alpha(y) + alpha(yx) = 0; for all x,y is an element of lambda.