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 withδ1 and δ2,thenδ1δ2(N) = 0implies
 δ1 =0orδ2 =0.