Some results on rings with chain conditions
type: Item
instance of: scholarly article
title: Some results on rings with chain conditions
zbMATH Open document ID: 0585.16006
DOI: 10.1007/BF01163608
author: Dinh Van Huynh
published in: Mathematische Zeitschrift
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/173655
review text: If A is a right noetherian ring such that A/P is right artinian for any prime ideal P, then A is right artinian if and only if its additive group has no direct summand isomorphic to the infinite cyclic group. A right noetherian ring is right artinian if and only if for any prime ideal P, A/P is a simple ring having a non-zero socle and for any homomorphic image A' of A the minimum condition holds for such principal right ideals of A' which are contained in the annihilator of A'. Also strictly linearly compact rings which are topologically noetherian, are characterised, and as a corollary the author gets that for any strictly linearly compact ring A the maximal divisible torsion ideal of A is contained in the left annihilator of A. If A is a strictly linearly compact ring such that the intersection of the closures of all finite powers of the Jacobson radical J is 0 and \(A/cl(J^ 2)\) is commutative, then A is algebraically and topologically isomorphic to a complete direct sum of strictly linearly compact local rings with identity and of a strictly linearly compact Jacobson radical ring.
Mathematics Subject Classification ID: 16P40
Mathematics Subject Classification ID: 16P20
Mathematics Subject Classification ID: 16D70
Mathematics Subject Classification ID: 16W80
zbMATH DE Number: 3937351
zbMATH Keywords: right noetherian ring
zbMATH Keywords: additive group
zbMATH Keywords: socle
zbMATH Keywords: minimum condition
zbMATH Keywords: principal right ideals
zbMATH Keywords: annihilator
zbMATH Keywords: strictly linearly compact rings
zbMATH Keywords: complete direct sum of strictly linearly compact local rings
zbMATH Keywords: Jacobson radical ring
reviewed by: R. Wiegandt
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3937351