Continuous rings with chain conditions
type: Item
instance of: scholarly article
title: Continuous rings with chain conditions
zbMATH Open document ID: 0824.16020
DOI: 10.1016/0022-4049(94)00017-4
published in: Journal of Pure and Applied Algebra
publication date: 1995-11-01T00:00:00Z
review text: A ring \(R\) is called left continuous if every left ideal of \(R\) is essential in a direct summand of \(R\) and every left ideal of \(R\) isomorphic to a direct summand of \(R\) is itself a direct summand of \(R\). Several interesting characterizations for left continuous rings to be quasi-Frobenius rings are given via the following conditions: \((\text{A}_ 1)\): ACC on left annihilators; \((\text{A}_ 2)\): \(R/\text{Soc} ({_ RR})\) is left Goldie; \((\text{S}_ 1)\): \(S = r(\ell(S))\) for every minimal right ideal \(S\), and \((\text{S}_ 2)\): Every minimal right ideal is essential in a direct summand of \(R_ R\). In fact, it is shown that the quasi-Frobenius rings are the left continuous rings satisfying either \((\text{A}_ 1)\) or \((\text{A}_ 2)\) and either \((\text{S}_ 1)\) or \((\text{S}_ 2)\). These characterizations extend several results [by \textit{P. Ara} and \textit{J. K. Park}, Commun. Algebra 19, 1945-1957 (1991; Zbl 0732.16018), \textit{E. P. Armendariz} and \textit{J. K. Park}, Arch. Math. 58, 24-33 (1992; Zbl 0768.16001), and \textit{V. Camillo} and \textit{M. F. Yousif}, Can. Math. Bull. 34, 462-464 (1991; Zbl 0767.16004)]. Furthermore, the following interesting fact on a certain class of left continuous rings is obtained: Assume that \(R\) is a left continuous ring such that \(R/ \text{Soc} ({_ RR})\) has ACC on left annihilators. If \(S = r(\ell (S))\) for every minimal right ideal \(S\) of \(R\), then \(\text{Soc} ({_ RR}) = \text{Soc} (R_ R)\), \(\text{Soc} (eR)\) is simple for every primitive idempotent \(e\) of \(R\) and there exists a complete set of distinct representatives \(\{Rt_ 1, \dots, Rt_ n\}\) of the isomorphism classes of the simple left \(R\)-modules such that \(\{t_ 1 R, \dots, t_ n R\}\) is a complete set of distinct representatives of the isomorphism classes of the simple right \(R\)-modules.
Mathematics Subject Classification ID: 16L60
Mathematics Subject Classification ID: 16P60
Mathematics Subject Classification ID: 16D60
Mathematics Subject Classification ID: 16E50
Mathematics Subject Classification ID: 16D70
zbMATH DE Number: 735695
zbMATH Keywords: direct summands
zbMATH Keywords: left continuous rings
zbMATH Keywords: quasi-Frobenius rings
zbMATH Keywords: ACC on left annihilators
zbMATH Keywords: left Goldie rings
zbMATH Keywords: minimal right ideals
zbMATH Keywords: primitive idempotents
zbMATH Keywords: simple left modules
author: W. Keith Nicholson
author: Mohamed F. Yousif
reviewed by: Jae Keol Park
MaRDI profile type: Publication
full work available at URL: 0022-4049(94)00017-4
OpenAlex ID: W1964278780
cites work: On continuous semiprimary rings
cites work: Self-injective rings with restricted chain conditions
cites work: Q3820710
cites work: Continuous Rings with Acc on Annihilators
cites work: CS-Modules with ACC or DCC on essential submodules
cites work: Continuous Rings with Acc on Essentials are Artinian
cites work: Q3674835
cites work: Q3995216
cites work: Q3199588
cites work: On Continuous Regular Rings and Semisimple Self Injective Rings
cites work: On Continuous Rings and Self Injective Rings
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description: scientific article; zbMATH DE number 735695