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# Article

 Title: On near-ring ideals with $(\sigma,\tau)$-derivation (English) Author: Golbaşi, Öznur Author: Aydin, Neşet Language: English Journal: Archivum Mathematicum ISSN: 0044-8753 (print) ISSN: 1212-5059 (online) Volume: 43 Issue: 2 Year: 2007 Pages: 87-92 Summary lang: English . Category: math . 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 $a\in Z$. (English) Keyword: prime near-ring Keyword: derivation Keyword: $(\sigma, \tau )$-derivation MSC: 16A70 MSC: 16A72 MSC: 16Y30 idZBL: Zbl 1156.16030 idMR: MR2336961 . Date available: 2008-06-06T22:50:41Z Last updated: 2012-05-10 Stable URL: http://hdl.handle.net/10338.dmlcz/108054 . Reference: [1] Bell H. E., Mason G.: On Derivations in near-rings, Near-rings and Near-fields.North-Holland Math. Stud. 137 (1987). MR 0890753 Reference: [2] Bell H. E.: On Derivations in Near-Rings II.Kluwer Acad. Publ., Dordrecht 426 (1997), 191–197. Zbl 0911.16026, MR 1492193 Reference: [3] Gölbaşi Ö., Aydin N.: Results on Prime Near-Rings with $(\sigma ,\tau )$-Derivation.Math. J. Okayama Univ. 46 (2004), 1–7. Zbl 1184.16049, MR 2109220 Reference: [4] Pilz G.: Near-rings.2nd Ed., North-Holland Math. Stud. 23 (1983). Zbl 0574.68051, MR 0721171 .

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