GENERALIZED DERIVATIONS WITH ANNIHILATOR CONDITIONS IN PRIME RINGS

Let R be a noncommutative prime ring with its Utumi ring of quotients U, C = Z(U) the extended centroid of R, F a generalized derivation of R and I a nonzero ideal of R. Suppose that there exists 0 not equal a is an element of R such that a(F([x, y])(n) - [x, y]) = 0 for all x, y is an element of I, where n >= 2 is a fixed integer. Then one of the following holds: 1. char (R) not equal 2, R subset of M-2(C), F(x) = bx for all x is an element of R with a(b - 1) = 0 (In this case n is an odd integer); 2. char (R) = 2, R subset of M-2(C) and F(x) = bx + [c, x] for all x is an element of R with a(b(n) - 1) = 0.