On (σ, τ)-derivations of Prime Near-Rings-II

Let N be a left near-ring and let σ, τ be automorphisms of N. An additive mapping d : N N is called a (σ, τ)-derivation on N if d(xy) = σ (x)d(y) + d(x)τ (y)~for all x,y ∈ N. In this paper, we obtain Leibniz' formula for (σ, τ)-derivations on near-rings which facilitates the proof of the following result: Let n ≥ 1 be an integer, N be n-torsion free, and d a (σ,τ)-derivation on N with dⁿ(N)=\0\. If both σ and τ commute with dⁿ for all n ≥ 1, then d(Z)=\0\. Further, besides proving some more related results, we investigate commutativity of N satisfying either of the properties: $d([x,y])= 0,$ ~or~ $d(xoy)= 0$, for all x,y ∈ N. 2000 Mathematics Subject Classification. 16W25, 16Y30