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 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 circle y)(m) - (x circle y)(n)) = 0 for all x, y is an element of I. Then one of the following 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 circle y)(m) = (x circle 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 circle y)(m) - (x circle 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)circle y(0))(m) - (x(0)circle 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 circle y)(m) - (x circle y)(n). is an element of Z(R) for any x, y is an element of R and there exist b is an element of Z(R) such that b(m) = 1.