Posner and herstein theorems for derivations of 3-prime near-rings

Abstract

The analog of Posner's theorem on the composition of two derivations in prime rings is proved for 3-prime near-rings. It is shown that if d is a nonzero derivation of a 2-torsionfree 3-prime near-ring N and an element a ∊ N is such that ax^d = x^da for all x ∊ N, then a is a central element. As a consequence it is shown that if d\ and d₂ are nonzero derivations of a 2-torsionfree 3-prime near-ring N and x^d1y^d2 = y^d2x^d1 for all x, y ∊ N, then N is a commutative ring. Thus two theorems of Herstein are generalized

1991 Mathematics Subject Classification:

• Primary 16Y30