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