On generalized (α,β)-derivations and Lie ideals of prime rings

Let R be a prime ring and alpha, beta be the automorphisms of R. The main aim of this article is to investigates several algebraic identities involving generalized (alpha, beta)-derivations acting on Lie ideals of prime rings. More precisely, we study the following identities: (i) F([x, y]) = alpha(x)circle F(y), (ii) F(x circle y) = [alpha(x), F(y)], (iii) F(xy) +/- yx. Z(R), (iv) F(xy) is an element of Z(R), (v) F(x)alpha (y) - beta(x)G(y) is an element of Z(R), (vi) F(x)y = xF(y), (vii) a(F(x)F(y) +/- alpha(xy)) = 0, (viii) a(F(x)F(y) +/- alpha(yx)) = 0 for all x, y. L (the nonzero square-closed Lie ideal of R), where 0. a. R is a fixed element. Moreover, some examples are given that exhibit the cruciality of the hypotheses taken.