COMMUTATIVITY OF PRIME NEAR RINGS

Abstract

The purpose of this paper is to study and generalize some results of [1] and [6] on commutativity in prime near-rings. Let N be a prime near-ring with multiplicative centre Z. Let σ and τ be automorphisms of N and δ be a (σ,τ)-derivation of N such that σ(δ(a)) = δ(σ(a)) and τ(δ(a)) = δ(τ(a)), for all a ∈ N. The following results are proved: 1. If N is 2-torsion free and δ(N) ⊆ Z, orδ(x)δ(y)=δ(y)δ(x), for all x,y ∈ N, then N is a commutative ring. 2. If N is 2-torsion free, δ1 is a derivation of N, δ2 is a (σ,τ)-derivation of N such that τ commutes with δ1 and δ2, thenδ1δ2(N) = 0 implies δ1 = 0 orδ2 = 0.