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

Archivum Mathematicum (2007)

- Volume: 043, Issue: 2, page 87-92
- ISSN: 0044-8753

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topGolbaşi, 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 -

## References

top- Bell H. E., Mason G., On Derivations in near-rings, Near-rings and Near-fields, North-Holland Math. Stud. 137 (1987). (1987) MR0890753
- Bell H. E., On Derivations in Near-Rings II, Kluwer Acad. Publ., Dordrecht 426 (1997), 191–197. (1997) Zbl0911.16026MR1492193
- Gölbaşi Ö., Aydin N., Results on Prime Near-Rings with $(\sigma ,\tau )$-Derivation, Math. J. Okayama Univ. 46 (2004), 1–7. Zbl1184.16049MR2109220
- Pilz G., Near-rings, 2nd Ed., North-Holland Math. Stud. 23 (1983). (1983) Zbl0574.68051MR0721171

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