Quasi-continuous modules relative to module classes
type: Item
instance of: scholarly article
title: Quasi-continuous modules relative to module classes
zbMATH Open document ID: 0949.16004
author: Semra Doǧruöz
author: Patrick F. Smith
published in: Vietnam Journal of Mathematics
publication date: 2000-09-11T00:00:00Z
review text: A module \(M\) is called quasi-continuous if \(f(M)\subseteq M\) for every idempotent endomorphism \(f\) of the injective envelope \(E(M)\) of \(M\). Quasi-continuous modules have been studied extensively, e.g., by \textit{S. H. Mohamed} and \textit{B. J. Müller} [Continuous and discrete modules, Lond. Math. Soc. Lect. Note Ser. 147, Cambridge Univ. Press (1990; Zbl 0701.16001)]. An equivalent characterization of a quasi-continuous module \(M\) is that for all submodules \(N_1\) and \(N_2\) of \(M\) with \(N_1\cap N_2=0\), there exist submodules \(M_1\) and \(M_2\) of \(M\) such that \(M=M_1\oplus M_2\) and \(N_i\subseteq M_i\) (\(i=1,2\)). Motivated by this, the authors investigate modules \(M\) satisfying the property that for each submodule \(N\) in some given class \(\mathcal X\) of modules and a submodule \(L\) with \(N\cap L=0\), there exist submodules \(N'\) and \(L'\) of \(M\) such that \(M=N'\oplus L'\) and \(N\subseteq N'\), \(L\subseteq L'\). The authors describe several general properties, and consider also special cases when \(\mathcal X\) is a specific class, e.g., the class of Goldie torsion modules. Note that this paper is closely related to earlier work of \textit{K. Oshiro} [Osaka J. Math. 20, 681-694 (1983; Zbl 0514.16016)], where quasi-continuous modules relative to a module class are also studied.
reviewed by: Nguyen Viet Dung
Mathematics Subject Classification ID: 16D70
Mathematics Subject Classification ID: 16D50
Mathematics Subject Classification ID: 16D80
zbMATH DE Number: 1417646
zbMATH Keywords: quasi-continuous modules
zbMATH Keywords: extending modules
zbMATH Keywords: \(M\)-injective modules
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1417646