On Generalized Left Derivation on Semiprime Rings


Let R be a 2-torsion free semiprime ring. If R admits a generalizedleft derivation F associated with Jordan left derivation d, then R is commutative, if any one of the following conditions hold: (1) [d(x), F(y)] [x, y], (2) [d(x), F(y)] xoy, (3) d(x)oF(y) xoy, (4) d(x)oF(y) [x, y], for all x, yR.