The Bass-Papp theorem and some related results
type: Item
instance of: scholarly article
title: The Bass-Papp theorem and some related results
zbMATH Open document ID: 0885.16007
published in: Vietnam Journal of Mathematics
publication date: 1998-04-06T00:00:00Z
review text: The well-known Bass-Papp Theorem states that a ring \(R\) is right Noetherian if and only if every direct sum of injective right \(R\)-modules is injective. A right \(R\)-module \(M\) is called a \(CS\)-module if every submodule of \(M\) is essential in a direct summand of \(M\). If \(M\) is a \(CS\)-module and for all direct summands \(K\) and \(L\) of \(M\) with \(K\cap L=0\), the submodule \(K\oplus L\) is also a direct summand of \(M\), then \(M\) is called quasi-continuous. \(CS\)-modules and quasi-continuous modules generalize injective modules, and the paper under review deals with the question of characterizing the rings over which direct sums of \(CS\)-modules (or quasi-continuous modules) are again \(CS\) (respectively, quasi-continuous). It is shown that the direct sum of any two quasi-continuous right \(R\)-modules is quasi-continuous if and only if \(R\) is semisimple Artinian. The similar question about when the class of \(CS\)-modules is closed under direct sums seems to be more difficult, and has no definitive answer in the paper. Among other partial results, it is shown that if the direct sum of any two \(CS\) right \(R\)-modules is \(CS\), then the ring \(R\) must be semi-artinian. Furthermore, if \(R\) has finitely generated right socle, then the direct sum of any two \(CS\) right \(R\)-modules is \(CS\) if and only if every right \(R\)-module is \(CS\), if and only if \(R\) is (left and right) Artinian serial and \(J(R)^2=0\).
reviewed by: Nguyen Viet Dung
Mathematics Subject Classification ID: 16D70
Mathematics Subject Classification ID: 16D50
Mathematics Subject Classification ID: 16P20
zbMATH DE Number: 1054656
zbMATH Keywords: injective right modules
zbMATH Keywords: direct summands
zbMATH Keywords: \(CS\)-modules
zbMATH Keywords: quasi-continuous modules
zbMATH Keywords: injective modules
zbMATH Keywords: direct sums
zbMATH Keywords: finitely generated right socle
author: Abdullah Harmanci
author: Patrick F. Smith
author: A. Tercan
author: Yücel Tiraş
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1054656