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On Minimal Varieties of Near-Rings

Chapter
Part of the Mathematics and Its Applications book series (MAIA, volume 336)

Abstract

It is well known that every minimal variety of associative rings is generated by a finite ring of prime order, in particular it is locally finite. In this paper we focus at locally finite minimal varieties of near-rings. They are exactly the varieties generated by finite strictly simple near-rings. We prove that every finite strictly simple near-ring is either a near-ring with the so-called trivial multiplication on a group of prime order or a finite planar near-ring whose additive group is elementary abelian. We describe the multiplicative subgroups of Galois fields which lead to strictly simple Ferrero near-rings and prove that in this way one obtains all finite strictly simple near-rings satisfying the identity xyz= yxz. In particular, this proves that the finite, strictly simple near-rings with non-prime order are abundant.

Keywords

Minimal Variety Prime Order Zero Multiplication Elementary Abelian Group Galois Field 
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Copyright information

© Springer Science+Business Media Dordrecht 1995

Authors and Affiliations

  1. 1.Department of MathematicsNational Cheng Kung UniversityTainan 701Taiwan, R.O.C.
  2. 2.Department of MathematicsTartu UniversityTartuEstonia
  3. 3.Department of MathematicsNational Cheng Kung UniversityTainan 701Taiwan, R.O.C

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