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Let R be a prime ring, I a nonzero ideal of R and n a fixed positive integer. If R admits a generalized derivation F associated with a derivation d such that (F ([x, y])) n = [x, y] for all x, y ∈ I. Then either R is commutative or n = 1, d = 0 and F is the identity map on R. Moreover in case R is a semiprime ring and (F ([x, y])) n = [x, y] for all x, y ∈ R, then either R is commutative or n = 1, d(R) ⊆ Z(R), R contains a non-zero central ideal and F (x) − x ∈ Z(R) for all x ∈ R.doi:10.4134/bkms.2011.48.6.1253 fatcat:2dbktewslvbnzf5f4vnc6ulr6y