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$\sigma$-derivations on prime near-rings. (English) Zbl 0992.16035
Let $N$ denote a zero-symmetric left near-ring and $\sigma$ an automorphism of $N$. An additive endomorphism $D$ of $N$ is called a $\sigma$-derivation if $D(xy)=\sigma(x)D(y)+D(x)y$ for all $x,y\in N$. This paper extends some commutativity results involving derivations, due to the reviewer and {\it G. Mason} [Near-rings and near-fields, Proc. Conf., Tübingen/F.R.G. 1985, North-Holland Math. Stud. 137, 31-35 (1987; Zbl 0619.16024)]. A typical theorem reads as follows: If $N$ is a 3-prime near-ring admitting a nontrivial $\sigma$-derivation $D$ such that $D(x)D(y)=D(y)D(x)$ for all $x,y\in N$, then $(N,+)$ is Abelian. Moreover, if $N$ is 2-torsion-free and $\sigma$ and $D$ commute, then $N$ is a commutative ring.

##### MSC:
 16Y30 Near-rings 16W25 Derivations, actions of Lie algebras (associative rings and algebras) 16U70 Center, normalizer (invariant elements) for associative rings 16U80 Generalizations of commutativity (associative rings and algebras) 16N60 Prime and semiprime associative rings 16W20 Automorphisms and endomorphisms of associative rings