On near-ring ideals with (?, ?)-derivation

Abstract

Let N be a 3-prime left near-ring with multiplicative center Z, a (?, ?)-derivation D on N is defined to be an additive endomorphism satisfying the product rule D(xy) = ?(x)D(y) + D(x)?(y) for all x, y ? N, where ? and ? are automorphisms of N. A nonempty subset U of N will be called a semigroup right ideal (resp. semigroup left ideal) if U N ? U (resp. NU ? U) and if U is both a semigroup right ideal and a semigroup left ideal, it be called a semigroup ideal. We prove the following results: Let D be a (?, ?)-derivation on N such that ?D = D?, ?D = D?. (i) If U is semigroup right ideal of N and D(U) ? Z then N is commutative ring, (ii) If U is a semigroup ideal of N and D2(U) = 0 then D = 0. (iii) If a ? N and [D(U),a]?? = 0 then D(a) = 0 or a ? Z.