| 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 |
| . |
