On Jacobson type radicals of near-rings
type: Item
instance of: scholarly article
title: On Jacobson type radicals of near-rings
zbMATH Open document ID: 0644.16027
DOI: 10.1007/BF01903365
author: Kalle Kaarli
published in: Acta Mathematica Hungarica
publication date: 1987-01-01T00:00:00Z
review text: In the following, all near-rings will be zero-symmetric left near-rings. In 1963 the reviewer introduced the Jacobson type radicals \(J_ 0\), \(J_ 1\), and \(J_ 2\) of a near-ring \(N\), where \(J_ i(N)\) is defined as the intersection of all annihilators of \(N\)-groups of type \(i\), \(i=0,1,2\). In a previous paper [Uch. Zap. Tartu Gos. Univ. 390, 134--171 (1976; Zbl 0397.16038)], the present author proved: a) \(J_ i\) is a Kurosh-Amitsur radical iff it is idempotent, i.e. iff \(J_ i(J_ i(N))=J_ i(\) number of edges, is called an optimal set. Similarly, if \(M\) and \(N\) are sets of \(k\) points, then \(N\) is said to be better than \(M\) if either \(f(N)>f(M)\), or \(f(N)=f(M)\) and the graph of \(N\) has fewer edges than the graph of \(M\). Let \(Q\) be a vertex of \(G\), and let \(P\) and \(R\) which are not neighbors of each other be neighbours of \(Q\). It is assumed that the angular region determined by \(QP\) and \(QR\) does not contain a neighbour of \(Q\). Let \(Q^*\) be the image of \(Q\) by reflection in the great circle \(PR\). \(M\) is said to be irreducible if it cannot be improved by a displacement or a single reflection \(Q\to Q^*\) of a single point \(Q\) of \(M\). Every optimal set is irreducible. Finally, let \(\phi_ 3(d)\) denote the interior angle of the equilateral triangle with side length \(d\). The main results of this very remarkable paper are the following: (i) For each \(k=7\) to \(12\) there is, up to isometry, exactly one optimal \(k\)-set \(M_ k\). \(M_{12}\) is the set of vertices of a regular icosahedron; \(M_{11}\) is obtained from \(M_{12}\) by removing one vertex. (ii) If \(M\) is irreducible and \(f(M)=d\) such that \(90\circ >\phi_ 3(d)\geq 72\circ,\) then \(k\leq 12\) and the graph of \(M\) is isomorphic to one of 21 specified graphs. Six of these sets are \(M_ 7,\cdots,M_{12}\). Ten of the remaining sets depend on one or two parameters, the exact ranges of which are explicitly given. The sets are uniquely determined (up to…
Mathematics Subject Classification ID: 16Y30
Mathematics Subject Classification ID: 16Nxx
zbMATH DE Number: 4049826
zbMATH Keywords: zero-symmetric left near-rings
zbMATH Keywords: Jacobson type radicals
zbMATH Keywords: Kurosh-Amitsur radical
zbMATH Keywords: optimal k-set
MaRDI profile type: Publication
cites work: Q5513080
cites work: Q4177729
cites work: Q4180504
cites work: Q3043338
cites work: Q4119339
cites work: Radicals for near-rings
description: scientific article; zbMATH DE number 4049826