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

Golbaşi, Öznur ; Aydin, Neşet

Archivum mathematicum, Tome 43 (2007) no. 2, pp. 87-92.

Résumé

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 $a\in Z$.

Détail

Classification : 16A70, 16A72, 16Y30
Keywords: prime near-ring; derivation; $(\sigma, \tau )$-derivation