Additive groups of trivial near-rings
type: Item
instance of: scholarly article
title: Additive groups of trivial near-rings
zbMATH Open document ID: 0848.16036
DOI: 10.1007/BF01874611
author: Shalom Feigelstock
published in: Acta Mathematica Hungarica
publication date: 1996-10-27T00:00:00Z
review text: A near-ring on a group \(G\) is trivial if there exists a subset \(S \subseteq G\) such that multiplication in \(G\) is defined by \(x.y = y\) if \(x \in S\), and is 0 otherwise. A group \(G\) is a TNR (trivial near-ring)-group if every near-ring on \(G\) is trivial. If \(|G|\leq 2\) then \(G\) is a TNR-group. Are there any others? The main theorem states that if \(G\) has an endomorphism, not the identity or zero which is either idempotent, or has square 0 or the identity, then \(G\) is not a TNR-group. Among other consequences the author shows that the only finite or abelian TNR-groups are those of order 1 or 2.
Mathematics Subject Classification ID: 16Y30
Mathematics Subject Classification ID: 20D45
Mathematics Subject Classification ID: 20K30
zbMATH DE Number: 878090
zbMATH Keywords: trivial near-rings
zbMATH Keywords: finite groups
zbMATH Keywords: trivial near-ring groups
zbMATH Keywords: multiplication
zbMATH Keywords: endomorphisms
zbMATH Keywords: abelian TNR-groups
reviewed by: John D. P. Meldrum
MaRDI profile type: Publication
cites work: Q4280394
cites work: Q3807388
description: scientific article; zbMATH DE number 878090