A Pair of Generalized Derivations in Prime, Semiprime Rings and in Banach Algebras

Let R be a prime ring with extended centroid C, I a non-zero ideal of R and n >= 1 a fixed integer. If R admits the generalized derivations H and G such that (H(xy) + G(yx))(n) = (xy +/- yx) for all x, y is an element of I, then one of the following holds: 1. R is commutative; 2. n = 1 and H(x) = x and G(x) = +/- x for all x is an element of R. Moreover, we examine the case where R is a semiprime ring. Finally, we apply the above result to non-commutative Banach algebras.