ON NEAR-RING IDEALS WITH (?,t)-DERIVATION

Let N be a 3-prime left near-ring with multiplicative center Z, a (sigma, tau)-derivation D on N is defined to be an additive endomorphism satisfying the product rule D (x y) = tau (x) D (y)+ D (x) sigma (y) for all x, y 2 N, where sigma and tau are automorphisms of N. A nonempty subset U of N will be called a semigroup right ideal (resp. semigroup left ideal) if U N subset of U (resp. N U subset of 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 (sigma, tau)-derivation on N such that sigma D = D sigma, tau D = D tau. (i) If U is semigroup right ideal of N and D (U) subset of Z then N is commutative ring. (ii) If U is a semigroup ideal of N and D (2) (U) = 0 then D = 0. (i i i) If a is an element of N and [D (U), a] sigma, tau = 0 then D (a) = 0 or a is an element of Z.