Some results by Bell and Mason on commutativity in near-rings are generalized. Let N be a prime right near-ring with multiplicative center Z and let D be a (σ, τ )-derivation on N such that σD = Dσ and τ D = Dτ . The following results are proved: (i) If D(N ) ⊂ Z or [D(N ), D(N )] = 0 or [D(N ), D(N )] σ,τ = 0 then (N, +) is abelian; (ii) If D(xy) = D(x)D(y) or D(xy) = D(y)D(x) for all x, y ∈ N then D = 0.

Abstract. In this paper, we extend the results concerning generalized derivations of prime rings in [2] and [8] for a nonzero Lie ideal of a prime ring R:

No abstract

Let R be a * −prime ring with characteristic not 2, U be a nonzero * −(σ, τ )−Lie ideal of R and d be a nonzero derivation of R. Suppose σ, τ be two automorphisms of R such that σd = dσ, τ d = dτ and * commutes with σ, τ, d. In the present paper it is shown that if d 2 (U ) = (0), then U ⊆ Z.

Let R be a prime ring with characteristic different from 2, let U be a nonzero Lie ideal of R, and let f be a generalized derivation associated with d. We prove the following results: (i) If a ∈ R and [a, f (U )] = 0 then a ∈ Z or d(a) = 0 or U ⊂ Z; (ii) If f 2 (U ) = 0 then U ⊂ Z; (iii) If u 2 ∈ U for all u ∈ U and f acts as a homomorphism or antihomomorphism on U then either d = 0 or U ⊂ Z.