Familiar examples of generalized derivations are derivations and generalized inner derivations, and the latter include left multipliers and right multipliers. Since the sum of two generalized derivations is a generalized derivation, every map of the form F(x) = xc +d(x), where c is fixed element of R and d is a derivation, is a generalized derivation; and if R has 1, all generalized derivations have this form. An additive subgroup L of R is said to be a Lie ideal of R if [L,R] ⊆ L. A Lie ideal L is said to be a square closed Lie ideal if x2 ∈ L for all x ∈ L. In [10] Posner showed that if a prime ring has a nontrivial derivation which is centralizing on the entire ring, then the ring must be commutative. Mayne [7] proved the same result for a prime ring with a nontrivial centralizing automorphism deals. A number of authors have generalized these results by considering mappings which are only assumed to be centralizing on an appropriate ideal of the ring. In [8] Mayne showed that if R is a prime ring with a nontrivial centralizing automorphism on a nonzero ideal or (quadratic) Jordan ideal, then R is commutative. For a prime ring with characteristic not equal two, an improved version was given in Mayne [9] by showing that a nontrivial automorphism which is centralizing on a Lie ideal implies that the ideal is contained in the center of the ring. In this paper, we shall show that L ⊆ Z(R) such that R admits a generalized derivations F and G with associated derivations d and g satisfying the following prope.