Minimal and maximal ideals in rings with involution
type: Item
instance of: scholarly article
title: Minimal and maximal ideals in rings with involution
zbMATH Open document ID: 0884.16021
author: G. F. Birkenmeier
author: Nico J. Groenewald
author: Henry E. Heatherly
published in: Beiträge zur Algebra und Geometrie
publication date: 1998-03-30T00:00:00Z
full work available at URL: https://eudml.org/doc/233795
full work available at URL: no.2
review text: Let \(R\) be a ring with involution *. If \(I\) is a *-minimal ideal of \(R\) such that \(I^2\neq 0\), then \(I^2=I\) and \(I\) is a *-simple ring. The socle of \(R\) is a direct sum of the sum of all *-minimal ideals and of the direct sum of minimal ideals \(K\) of \(R\) such that \(K^2=0\) and \(K\oplus K^*\) is not a *-minimal ideal of \(R\). *-maximal ideals of a ring with involution are investigated and described in terms of maximal ideals, in particular, if \(R^2=R\) and \(M\) is a maximal ideal of \(R\), then \(M\cap M^*\) is a *-maximal ideal of \(R\). In a ring with involution every proper *-prime ideal is a *-maximal ideal if and only if every proper prime ideal is maximal.
reviewed by: R. Wiegandt
Mathematics Subject Classification ID: 16W10
Mathematics Subject Classification ID: 16D25
zbMATH DE Number: 1066093
zbMATH Keywords: rings with involution
zbMATH Keywords: socles
zbMATH Keywords: *-minimal ideals
zbMATH Keywords: direct sums of minimal ideals
zbMATH Keywords: maximal ideals
zbMATH Keywords: prime ideals
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1066093