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

Oznur Golbasi, Neset Aydin

O. Golbasi, Cumhuriyet University, Faculty of Arts and Science, Department of Mathematics, Sivas - Turkey
N. Aydin,Canakkale 18 Mart University, Faculty of Arts and Science\newline Department of Mathematics, Canakkale - Turkey


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 \ $a\in Z$.

16A72, 16A70,16Y30.

Prime near-ring, derivation, $(\sigma,\tau)$-derivation.