Matrix near-rings and 0-primitivity.
type: Item
instance of: scholarly article
title: Matrix near-rings and 0-primitivity.
zbMATH Open document ID: 1244.16037
author: Wen-Fong Ke
author: Johan Meyer
published in: Monatshefte für Mathematik
publication date: 2012-03-23T00:00:00Z
review text: As for rings, the relationships between the radicals of a near-ring and those of the corresponding matrix near-rings have been thoroughly investigated. An intriguing remaining open question is whether the underlying base near-ring of a 0-primitive matrix near-ring must also be \(0\)-primitive. It is already known that the matrix near-ring over a \(0\)-primitive near-ring must be \(0\)-primitive and if the base near-ring is finite, the converse also holds. The authors do not answer the open question here, but they show that it has a positive answer under suitable finiteness conditions: If the \(0\)-primitive matrix near-ring \(\mathbb M_n(R)\) has the dcc on \(\mathbb M_n(R)\)-subgroups, then the base near-ring \(R\) has the dcc on \(R\)-subgroups and is also \(0\)-primitive. Any optimisism of a positive answer in general is slightly dampened by an example provided in the last section where a \(0\)-symmetric generalized matrix near-ring \(\mathbb M_2(R,G)\) is given over a base near-ring \(R\) which is not \(0\)-primitive.
reviewed by: Stefan Veldsman
Mathematics Subject Classification ID: 16Y30
Mathematics Subject Classification ID: 16S50
Mathematics Subject Classification ID: 16D60
Mathematics Subject Classification ID: 16N80
Mathematics Subject Classification ID: 16N20
zbMATH DE Number: 6018271
zbMATH Keywords: primitive near-rings
zbMATH Keywords: matrix near-rings
MaRDI profile type: Publication
full work available at URL: s00605-010-0267-z
OpenAlex ID: W2090914015
cites work: Matrix near rings
cites work: Left ideals in matrix near-rings
cites work: Left ideals and 0-primitivity in matrix near-rings
cites work: Q3671335
cites work: Generalized matrix near-rings
cites work: PRIMITIVITY IN MATRIX NEAR-RINGS
DOI: 10.1007/S00605-010-0267-Z
description: scientific article; zbMATH DE number 6018271