POWER VALUES OF GENERALIZED DERIVATIONS WITH ANNIHILATOR CONDITIONS IN PRIME RINGS

Let R be a prime ring, Q its symmetric Martindale quotient ring, C its extended centroid, I a nonzero ideal of R, and F a generalized derivation with associated derivation d of R, m >= 1, n >= 1 two fixed integers, and 0 not equal a is an element of R. 1. Assume that a ((F([x, y]))(m) - [x, y](n)) = 0, for all x, y is an element of I, then one of the following statements holds: (a) R is commutative; (b) n = m = 1 and there exists b is an element of Q such that F(x) = bx for all x is an element of R with ab= a; (c) There exists b is an element of C such that F(x) = bx for all x is an element of R with b(m) = 1 and [x, y](m) = [x, y](n), for all x, y is an element of R; d) R subset of M-2(C), the ring of 2 x 2 matrices over C; n = 1 and m >= 2 such that alpha(m) = alpha for all alpha is an element of C; and there exists b is an element of Q such that F(x) = bx for all x is an element of R with ab = a; (e) R subset of M-2(C) and char (R) = 2. 2. Assume that char (R) not equal 2 and a((F([x, y]))(m) - [x, y](n)) is an element of Z(R) for all x, y is an element of I. If there exist x(0), y(0) is an element of I such that a((F([x(0), y(0)]))(m) - [x(0), y(0)](n)) not equal 0, then either there exists a field E such that R subset of M-2(E) or a is an element of Z(R), [x, y](m) - [x, y](n) is an element of Z(R) for any x, y is an element of R, and there exist b, c is an element of Z(R) sucht that (b + c)(m) = 1.