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.
There is an increasing body of evidence that prime near-rings with derivations have ring like behavior, indeed, there are several results (see for example [1], [2], [3], [4], [5] and [8]) asserting that the existence of a suitably-constrained derivation on a prime near-ring forces the near-ring to be a ring. It is our purpose to explore further this ring like behaviour. In this paper we generalize some of the results due to Bell and Mason [4] on near-rings admitting a special type of derivation...
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