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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 ∈ 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 ∈ R. Moreover, we examine the case where R is a semiprime ring. Finally, we apply the above result to non-commutative Banach algebras.

A pair of generalized derivations in prime, semiprime rings and in Banach algebras