Commutativity Results on Prime Rings With Generalized Derivations


Let R be a prime ring. For nonzero generalized derivations F and G associated with the same derivation d, we prove that if d≠0, then R is commutative, if any one of the following conditions hold: (1) [F(x), G(y)]  0, (2) F(x)oG(y)  0, (3) F(x)oG(y)  xoy, (4) [F(x), G(y)]  [x, y], (5) [F(x), G(y)]  xoy, (6) F(x)oG(y)  [x, y], for all x, y  R, where F will always denote onto map.