On Minimal Varieties of Near-Rings
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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.
KeywordsMinimal Variety Prime Order Zero Multiplication Elementary Abelian Group Galois Field
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