Commutativity results on prime rings with generalized derivations

الملخص EN

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.