On essential extensions of reduced rings and domains.
type: Item
instance of: scholarly article
title: On essential extensions of reduced rings and domains.
zbMATH Open document ID: 1065.16004
published in: Archiv der Mathematik
publication date: 2005-01-20T00:00:00Z
review text: A ring is a left essential extension of a reduced ring (domain) if it contains a left ideal which is a reduced ring (domain) and intersects nontrivially every nonzero twosided ideal of the ring. It is known that if \(I\) is a reduced ring which is an essential ideal of a ring \(R\), then \(R\) itself is a reduced ring. The corresponding property for left essential extensions is known not to be true. This raises the question to describe the left essential extensions of reduced rings and domains which is addressed in this paper. The main result shows that any left essential extension of a reduced ring is a subdirect sum of rings which are left essential extensions of domains. This is achieved by firstly describing the relationship between minimal prime ideals of reduced rings and their left essential extensions. It is also shown that not all subdirect sums of rings which are left essential extensions of domains are left essential extensions of reduced rings. As an application of their results, it is shown that the upper radical class determined by the class of left essential extensions of domains is a special radical which is right but not left strong. Examples of such radical classes are known to exist, but they are relatively scarce.
reviewed by: Stefan Veldsman
Mathematics Subject Classification ID: 16D80
Mathematics Subject Classification ID: 16U10
Mathematics Subject Classification ID: 16N80
Mathematics Subject Classification ID: 16D25
Mathematics Subject Classification ID: 16D70
zbMATH DE Number: 2130532
zbMATH Keywords: left essential extensions
zbMATH Keywords: reduced rings
zbMATH Keywords: domains
zbMATH Keywords: right strong radicals
zbMATH Keywords: left essential ideals
zbMATH Keywords: subdirect sums of rings
zbMATH Keywords: minimal prime ideals
zbMATH Keywords: upper radical classes
zbMATH Keywords: special radicals
author: K. I. Beidar
author: Y. Fong
author: E. R. Puczyłowski
MaRDI profile type: Publication
full work available at URL: s00013-003-4786-x
OpenAlex ID: W1978400008
DOI: 10.1007/S00013-003-4786-X
description: scientific article; zbMATH DE number 2130532