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On near-ring ideals with $$(\sigma,\tau)$$-derivation. (English) Zbl 1156.16030
Summary: 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 $$D(xy)=\tau(x)D(y)+D(x)\sigma(y)$$ for all $$x,y\in N$$, 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 $$UN\subset U$$ (resp. $$NU\subset U$$) 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 a semigroup right ideal of $$N$$ and $$D(U)\subset Z$$, then $$N$$ is a commutative ring. (ii) If $$U$$ is a semigroup ideal of $$N$$ and $$D^2(U)=0$$ then $$D=0$$. (iii) If $$a\in N$$ and $$[D(U),a]_{\sigma,\tau}=0$$ then $$D(a)=0$$ or $$a\in Z$$.
##### MSC:
 16Y30 Near-rings 16W25 Derivations, actions of Lie algebras 16U70 Center, normalizer (invariant elements) (associative rings and algebras)
##### Keywords:
prime near-rings; derivations; commutativity theorems
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