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Zbl 0992.16035
Kamal, Ahmed A.M.
$\sigma$-derivations on prime near-rings. (English)
[J] Tamkang J. Math. 32, No.2, 89-93 (2001). ISSN 0049-2930; ISSN 2073-9826/e

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.
[Howard E.Bell (St.Catharines)]

MSC 2000:: *16Y30 Near-rings
16W25 Derivations, actions of Lie algebras (assoc. rings and algebras)
16U70 Commutativity theorems for assoc. rings
16U80 Generalizations of commutativity (assoc. rings and algebras)
16N60 Prime and semiprime assoc. rings
16W20 Morphisms of associative rings

Keywords: zero-symmetric left near-rings; automorphisms; additive endomorphisms; derivations; commutativity theorems

Citations: Zbl 0619.16024

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