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 $a\in Z$.
@article{108054,
author = {\"Oznur Golba\c si and Ne\c set Aydin},
title = {On near-ring ideals with $(\sigma,\tau)$-derivation},
journal = {Archivum Mathematicum},
volume = {043},
year = {2007},
pages = {87-92},
zbl = {1156.16030},
mrnumber = {2336961},
language = {en},
url = {http://dml.mathdoc.fr/item/108054}
}
Golbaşi, Öznur; Aydin, Neşet. On near-ring ideals with $(\sigma,\tau)$-derivation. Archivum Mathematicum, Tome 043 (2007) pp. 87-92. http://gdmltest.u-ga.fr/item/108054/
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