Some Conditions under Which Prime near-Rings Are Commutative Rings

Abstract

Let N be a prime left near-ring, and I be a nonzero semigroup ideal of N . We prove that if N admits a derivation d satisfying any one of the following properties: (i)d([x;y]) = [x;y], (ii)d([x;y]) = [d(x);y], (iii) [d(x);y] = [x;y], (iv)d(x y) = x y, (v)d(x) y = x y and (vi)d(x) y = x y for all x;y2 I; then N is a commutative ring. Moreover, example proving the necessity of the primeness condition is given.