Address.
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
E-mail.
ogolbasi@cumhuriyet.edu.tr
neseta@comu.edu.tr
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 \ $a\in Z$.
AMSclassification.
16A72, 16A70,16Y30.
Keywords.
Prime near-ring, derivation, $(\sigma,\tau)$-derivation.