On generalized left derivation on semiprime rings

الملخص EN

Let R be a 2-torsion free semiprime ring.

If R admits a generalized left 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)] oy, (3) d(x)oF(y) xoy, (4) d(x)oF(y) [x, y], for all x, y  R.