On almost QF rings
type: Item
instance of: scholarly article
title: On almost QF rings
zbMATH Open document ID: 0879.16007
author: Takeshi Sumioka
author: Shojiro Tozaki
published in: Osaka Journal of Mathematics
publication date: 1997-11-30T00:00:00Z
review text: A semiprimary ring \(R\) is called a right almost QF ring if \(R\) satisfies acc on right annihilators and every indecomposable projective right \(R\)-module is almost injective, and dually \(R\) is called a right almost co-QF ring if \(R\) satisfies dcc on right annihilators and every indecomposable injective right \(R\)-module is almost projective. Furthermore a semiprimary ring \(R\) is called a right \(H\) ring if \(R\) satisfies dcc on right annihilators and for a given non-small module \(M\), there exists a monomorphism from some nonzero injective module to \(M\). Dually a semiprimary ring \(R\) is called a right co-\(H\) ring if \(R\) satisfies acc on right annihilators and for a given non-cosmall module, there exists an epimorphism from \(M\) to some nonzero projective module. Without assuming that \(R\) is two-sided Artinian, it is shown that the following conditions are equivalent: (i) \(R\) is a right almost QF ring; (ii) \(R\) is a left almost co-QF ring; (iii) \(R\) is a right co-\(H\) ring; (iv) \(R\) is a left \(H\) ring. Also it is proved that under these conditions, the ring \(R\) is necessarily left Artinian.
Mathematics Subject Classification ID: 16L60
Mathematics Subject Classification ID: 16P60
zbMATH DE Number: 1018457
zbMATH Keywords: right almost QF rings
zbMATH Keywords: acc on right annihilators
zbMATH Keywords: indecomposable projective right modules
zbMATH Keywords: right almost co-QF rings
zbMATH Keywords: dcc on right annihilators
zbMATH Keywords: indecomposable injective right modules
zbMATH Keywords: right \(H\) rings
zbMATH Keywords: right co-\(H\) rings
reviewed by: Jae Keol Park
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1018457