Some properties of prime near-rings with (σ,τ)-derivation

Abstract

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 τD = 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.