On the cardinality of ideals in Artinian rings
type: Item
instance of: scholarly article
title: On the cardinality of ideals in Artinian rings
zbMATH Open document ID: 0658.16010
DOI: 10.1007/BF01207473
author: Dinh Van Huynh
author: Nguyen Viet Dung
published in: Archiv der Mathematik
publication date: 1988-01-01T00:00:00Z
review text: Let \(R\) be a right Artinian ring and let \(\alpha\) be an infinite cardinal number. The authors' main result shows that \(R\) contains two uniquely determined (two-sided) ideals \(S(\alpha)\) and \(S^*(\alpha)\) such that: (i) \(S(\alpha)\) is the largest right ideal of \(R\) with cardinality less than \(\alpha\), (ii) \(S^*(\alpha)\) is the smallest ideal of \(R\) such that \(R/S^*(\alpha)\) has cardinality less than \(\alpha\). Moreover, \(S^*(\alpha)\) has a right identity, (so that it is also right Artinian) and \(S^*(\alpha)\cap S(\alpha)=0\). Also, letting \(T(\alpha)\) be the sum of all the left ideals of \(R\) with cardinality less than \(\alpha\), if \(\text{card}\,T(\alpha)<\alpha\) (as is the case if \(R\) is also left Artinian) then \(T(\alpha)=S(\alpha)\) and \(R\) is the direct sum of \(S(\alpha)\) and \(S^*(\alpha)\). The authors show by example that this cardinality condition on \(T(\alpha)\) can not be omitted. As a consequence of these results the authors obtain several results on the structure of right Artinian rings. For example, they show that if \(R\) is right Artinian such that no two ideals of \(R\) have the same cardinality then \(R\) is the direct sum of a semiprime Artinian ring and a finite ring.
Mathematics Subject Classification ID: 16P20
Mathematics Subject Classification ID: 16Dxx
Mathematics Subject Classification ID: 16D70
zbMATH DE Number: 4075312
zbMATH Keywords: infinite cardinal number
zbMATH Keywords: left ideals
zbMATH Keywords: right Artinian rings
zbMATH Keywords: direct sum
zbMATH Keywords: semiprime Artinian ring
reviewed by: John Clark
MaRDI profile type: Publication
cites work: A note on Artinian rings
cites work: Some characterizations of hereditarily artinian rings
cites work: A characterization of artinian rings
cites work: The maximal regular ideal of self-injective and continuous rings splits off
cites work: Q5611865
cites work: A Countable Self-Injective Ring is Quasi-Frobenius
description: scientific article; zbMATH DE number 4075312