Self-duality of quasi-Harada rings and locally distributive rings.
type: Item
instance of: scholarly article
title: Self-duality of quasi-Harada rings and locally distributive rings.
zbMATH Open document ID: 1113.16013
author: Kazutoshi Koike
published in: Journal of Algebra
publication date: 2006-12-07T00:00:00Z
review text: Left quasi-Harada rings are studied (i.e., left Artinian rings \(R\) such that every finitely generated projective right \(R\)-module is quasi-injective), and conditions when such rings have a self-duality are indicated. The main results are the following: -- 1. Let \(R\) be a ring and let \(R'\) be a diagonally complete subring of \(R\). If \(R\) has a left or a right Morita duality, then so does \(R'\). If \(R\) has a good self-duality, then so does \(R'\). -- 2. If \(R\) is a left quasi-Harada ring, then \(R\) has a right Morita duality, in particular, \(R\) is right Artinian. -- 3. Every left quasi-Harada ring can be constructed from a QF-ring. -- 4. If \(R\) is a left quasi-Harada ring, then the existence of a good self-duality for \(R\), or \(Q^r_{\max}(R)\), or \(\Lambda(R)=\text{End}_R(ER_R))\) or \(\Gamma(R)=fRf\) are equivalent conditions. -- 5. Let \(R\) be a left quasi-Harada QF-3 ring with minimal faithful right \(R\)-module \(gR\), where \(g^2=g\in R\). If \(gRg\) is a serial ring, then \(R\) has a good self-duality (an improvement of a result of \textit{Y. Baba} [Commun. Algebra 30, No. 6, 2583-2592 (2002; Zbl 1008.16004)]). -- 6. The existence of a good self-duality for a left quasi-Harada ring can be reduced to that for a QF-ring. -- 7. If \(R\) is a left quasi-Harada ring with finite ideal lattice, then \(R\) has an almost self-duality. -- 8. Every locally distributive right QF-2 ring has an almost self-duality. Every locally distributive serial ring has a good self-duality.
reviewed by: A. I. Kashu
Mathematics Subject Classification ID: 16D90
Mathematics Subject Classification ID: 16P20
Mathematics Subject Classification ID: 16L60
zbMATH DE Number: 5078140
zbMATH Keywords: Morita dualities
zbMATH Keywords: QF-rings
zbMATH Keywords: left quasi-Harada rings
zbMATH Keywords: left Artinian rings
zbMATH Keywords: finitely generated projective right modules
zbMATH Keywords: self-dualities
zbMATH Keywords: locally distributive right QF-2 rings
zbMATH Keywords: serial rings
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2006.04.023
OpenAlex ID: W2026679833
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DOI: 10.1016/J.JALGEBRA.2006.04.023
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description: scientific article; zbMATH DE number 5078140