On Lie Structure of Prime Rings with Generalized (\alpha - doi: 10.5269/bspm.v27i2.10209

Abstract

Let R be a ring and , 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 ∈ R. For any x, y ∈ 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, y]) = (xoy)_{\alpha,\beta}, (ii) F(xoy) = [x, y]_{\alpha,\beta}, (iii) [F(x), y]_{\alpha,\beta} = (F(x)oy)_{\alpha,\beta}, (iv) F([x, y]) = [F(x), y]_{\alpha,\beta}, (v) F(xoy) = (F(x) o y)_{\alpha,\beta}, (vi) F([x, y] =[\alpha(x),G(y)] and (vii) F(xoy) = (\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.