Geodesic
On near-ring ideals with $(\sigma,\tau)$-derivation
Archivum mathematicum, Tome 43 (2007) no. 2, pp. 87-92 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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 $a\in Z$.
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 $a\in Z$.
Classification : 16A70, 16A72, 16Y30
Keywords: prime near-ring; derivation; $(\sigma, \tau )$-derivation 
@article{ARM_2007_43_2_a0,
     author = {Golba\c{s}i,  \"Oznur and Aydin, Ne\c{s}et},
     title = {On near-ring ideals with $(\sigma,\tau)$-derivation},
     journal = {Archivum mathematicum},
     pages = {87--92},
     year = {2007},
     volume = {43},
     number = {2},
     mrnumber = {2336961},
     zbl = {1156.16030},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ARM_2007_43_2_a0/}
}
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Golbaşi,  Öznur; Aydin, Neşet. On near-ring ideals with $(\sigma,\tau)$-derivation. Archivum mathematicum, Tome 43 (2007) no. 2, pp. 87-92. http://geodesic.mathdoc.fr/item/ARM_2007_43_2_a0/