ON COMMUTATIVITY OF PRIME RINGS WITH LEFT GENERALIZED DERIVATIONS.

Let R be a prime ring with center Z(R), central closure RC, right Martindale quotient ring Q_r, F a left generalized derivation and I a nonzero right ideal of R which is semiprime as a ring. We proved in this article that if one of the following conditions holds: (i) F(x o y) ± x o y = 0 (ii) F(xy) ± xy ∈ Z(R) (iii) F(x)F(y) ± xy ∈ Z(R) (iv) F([x,y]) = ±[F(x), y] (v) F(x o y) = ±(F(x) o y) for all x, y ∈ I, then R is commutative or there exists q ∈Q_r(RC) such that F(x) = qx for all x ∈ R. [ABSTRACT FROM AUTHOR]