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

Archivum Mathematicum (2007)

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

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## Abstract

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Let $N$ be a $3$-prime left near-ring with multiplicative center $Z$, a $\left(\sigma ,\tau \right)$-derivation $D$ on $N$ is defined to be an additive endomorphism satisfying the product rule $D\left(xy\right)=\tau \left(x\right)D\left(y\right)+D\left(x\right)\sigma \left(y\right)$ 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 $\left(\sigma ,$$\tau \right)$-derivation on $N$ such that $\sigma D=D\sigma ,\tau D=D\tau$. (i) If $U$ is semigroup right ideal of $N$ and $D\left(U\right)\subset Z$ then $N$ is commutative ring. (ii) If $U$ is a semigroup ideal of $N$ and ${D}^{2}\left(U\right)=0$ then $D=0.$ (iii) If $a\in N$ and ${\left[D\left(U\right),a\right]}_{\sigma ,\tau }=0$ then $D\left(a\right)=0$ or $a\in Z$.

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