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Keywords:
prime near-ring; derivation; $(\sigma, \tau )$-derivation
prime near-ring; derivation; $(\sigma, \tau )$-derivation
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 semigroup right ideal of $N$ and $D(U)\subset Z$ then $N$ is 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 $a\in Z$.
References:
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[2] Bell H. E.: On Derivations in Near-Rings II. Kluwer Acad. Publ., Dordrecht 426 (1997), 191–197. MR 1492193 | Zbl 0911.16026
[3] Gölbaşi Ö., Aydin N.: Results on Prime Near-Rings with $(\sigma ,\tau )$-Derivation. Math. J. Okayama Univ. 46 (2004), 1–7. MR 2109220 | Zbl 1184.16049
[4] Pilz G.: Near-rings. 2nd Ed., North-Holland Math. Stud. 23 (1983). MR 0721171 | Zbl 0574.68051
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