Minimal non-totally minimal topological rings
type: Item
instance of: scholarly article
title: Minimal non-totally minimal topological rings
zbMATH Open document ID: 0886.16031
published in: Rendiconti del Seminario Matematico della Università di Padova
publication date: 1998-05-10T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/108431
review text: Let \(R\) be a discrete ring and \(R^*\) the group of characters of its additive group. The group \(R^*\) can be regarded in a natural way with a structure of a topological \((R,R)\)-bimodule. The topological group \(R\times R^*\times(\mathbb{R}/\mathbb{Z})\) with the multiplication \((r_1,\xi_1,s_1)(r_2,\xi_2,s_2)=(r_1r_2,r_1\xi_2+\xi_1r_2,\xi_1(r_2)+\xi_2(r_1))\), where \(r_1,r_2\in R\), \(\xi_1,\xi_2\in R^*\), \(s_1,s_2\in\mathbb{R}/\mathbb{Z}\) becomes a locally compact associative ring (denoted by \(M(R)\)). The main result of the paper states that the topological ring \(M(R)\) is minimal. This result implies that there exists a locally compact minimal ring \(R\) with identity having a closed ideal \(I\) such that the factor ring \(R/I\) is not minimal. The last example gives a positive answer to a question posed by D. Dikranjan.
Mathematics Subject Classification ID: 16W80
Mathematics Subject Classification ID: 54H13
Mathematics Subject Classification ID: 54A10
Mathematics Subject Classification ID: 22B05
zbMATH DE Number: 1089771
zbMATH Keywords: groups of characters
zbMATH Keywords: topological groups
zbMATH Keywords: locally compact associative rings
zbMATH Keywords: locally compact minimal rings
author: Michael Megrelishvili
reviewed by: Mihail I. Ursul
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1089771