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On prime near-rings with generalized derivation. (English) Zbl 1152.16036

A generalized derivation \(f\) with associated derivation \(D\) in a (left) nearring \(N\) is an additive endomorphism of \(N\) such that \(f(xy)=f(x)y+xD(y)\), \(\forall x,y\in N\). Now \(3\)-prime \(2\)-torsion free nearrings with a generalized derivation fulfilling certain conditions are commutative rings.

MSC:

16Y30 Near-rings
16W25 Derivations, actions of Lie algebras
16U70 Center, normalizer (invariant elements) (associative rings and algebras)
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References:

[1] H. E. Bell and G. Mason, “On derivations in near-rings,” in Near-Rings and Near-Fields (Tübingen, 1985), G. Betsch, Ed., vol. 137 of North-Holland Mathematics Studies, pp. 31-35, North-Holland, Amsterdam, The Netherlands, 1987. · Zbl 0619.16024
[2] X.-K. Wang, “Derivations in prime near-rings,” Proceedings of the American Mathematical Society, vol. 121, no. 2, pp. 361-366, 1994. · Zbl 0811.16040 · doi:10.2307/2160409
[3] H. E. Bell and N.-U. Rehman, “Generalized derivations with commutativity and anti-commutativity conditions,” Mathematical Journal of Okayama University, vol. 49, no. 1, pp. 139-147, 2007. · Zbl 1139.16020
[4] Ö. Gölba\csi, “Notes on prime near-rings with generalized derivation,” Southeast Asian Bulletin of Mathematics, vol. 30, no. 1, pp. 49-54, 2006. · Zbl 1117.16033
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