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Let R be a prime ring with extended centroid C, F a generalized derivation of R and n ≥ 1, m≥ 1 fixed integers. In this paper we study the situations:
1. for all x,y ∈ I, where I is a nonzero ideal of R;
2. (F(x∘y))ⁿ=(x∘y)ⁿ for all x,y ∈ I, where I is a nonzero right ideal of R.
Moreover, we also investigate the situation in semiprime rings and Banach algebras.
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