On Semiprime Rings with Generalized Derivations

Abstract

Let R be a ring and F and G be generalized derivations of R with associated derivations d and g respectively. In the present paper,we shall investigate the commutativity of semiprime ring R admitting generalized derivations F and G satisfying any one of the properties: (i)F(x)x = xG(x), (ii) [F(x), d(y)] = [x,y], (iii) F([x,y]) = [d(x), F(y)], (iv) d(x)F(y) = xy, (v) F(x^2) = x^2, (vi) [F(x),y] = [x,G(y)], (vii) F([x, y]) = [F(x), y] + [d(y), x] and (viii) F(x ◦ y) = F(x) ◦ y −d(y) ◦ x for all x, y in some appropriate subset of R.

Keywords

• Ideal
• semiprime ring
• derivation
• generalized derivation.