On near-ring ideals with $(\sigma ,\tau )$-derivation

Golbaşi, Öznur, and Aydin, Neşet. "On near-ring ideals with $(\sigma ,\tau )$-derivation." Archivum Mathematicum 043.2 (2007): 87-92. <http://eudml.org/doc/250148>.

@article{Golbaşi2007,
abstract = {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 $a\in Z$.},
author = {Golbaşi, Öznur, Aydin, Neşet},
journal = {Archivum Mathematicum},
keywords = {prime near-ring; derivation; $(\sigma , \tau )$-derivation; prime near-rings; derivations; commutativity theorems},
language = {eng},
number = {2},
pages = {87-92},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {On near-ring ideals with $(\sigma ,\tau )$-derivation},
url = {http://eudml.org/doc/250148},
volume = {043},
year = {2007},
}

TY - JOUR
AU - Golbaşi, Öznur
AU - Aydin, Neşet
TI - On near-ring ideals with $(\sigma ,\tau )$-derivation
JO - Archivum Mathematicum
PY - 2007
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 043
IS - 2
SP - 87
EP - 92
AB - 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 $a\in Z$.
LA - eng
KW - prime near-ring; derivation; $(\sigma , \tau )$-derivation; prime near-rings; derivations; commutativity theorems
UR - http://eudml.org/doc/250148
ER -