Weak Krull-Schmidt for infinite direct sums of uniserial modules
type: Item
instance of: scholarly article
title: Weak Krull-Schmidt for infinite direct sums of uniserial modules
zbMATH Open document ID: 0885.16008
author: Nguyen Viet Dung
author: Alberto Facchini
published in: Journal of Algebra
publication date: 1998-04-06T00:00:00Z
review text: The purpose of this paper is to determine necessary and sufficient conditions for two arbitrary families \(\{U_i:i\in I\}\) and \(\{V_j:j\in J\}\) of uniserial modules to have the property \(\bigoplus_{i\in I} U_i\cong\bigoplus_{j\in J} V_j\). Most of the results proved by the second author for finite families of uniserial modules extend to arbitrary families [see \textit{A. Facchini}, Trans. Am. Math. Soc. 348, No. 11, 4561-4575 (1996; Zbl 0868.16003)]. The main new idea is that of a quasismall module: a module \(U\) is quasismall if whenever \(U\) is isomorphic to a direct summand of a direct sum \(\bigoplus_{\lambda\in\Lambda}M_\lambda\) of arbitrary modules \(M_\lambda\) (\(\lambda\in\Lambda\)), there is a finite subset \(F\subseteq\Lambda\) such that \(U\) is isomorphic to a direct summand of \(\bigoplus_{\lambda\in F} M_\lambda\). If \(U_i\) (\(i\in I\)) and \(V_j\) (\(j\in J\)) are all quasismall uniserial modules then a complete analogue of Facchini's Theorem is obtained. The class of quasismall uniserial modules includes all finitely generated and all uncountably generated uniserial modules and also all uniserial modules with local endomorphism ring. Another analogue of Facchini's Theorem is obtained in case \(\{U_i:i\in I\}\) and \(\{V_j:j\in J\}\) are locally semi-\(T\)-nilpotent families.
Mathematics Subject Classification ID: 16D70
Mathematics Subject Classification ID: 16P60
zbMATH DE Number: 1051515
zbMATH Keywords: semi-\(T\)-nilpotent families
zbMATH Keywords: uniserial modules
zbMATH Keywords: quasismall modules
zbMATH Keywords: direct summands
zbMATH Keywords: direct sums
zbMATH Keywords: local endomorphism rings
reviewed by: Patrick F. Smith
MaRDI profile type: Publication
full work available at URL: jabr.1996.6977
OpenAlex ID: W2016179255
cites work: Corrections and Supplementaries to My Paper concerning Krull-Remak-Schmidt’s Theorem
cites work: Refinements for infinite direct decompositions of algebraic systems
cites work: Krull-Schmidt fails for serial modules
cites work: Krull-Schmidt Fails for Artinian Modules
cites work: Uniserial modules over valuation rings
cites work: Krull-Schmidt uniqueness fails dramatically over subrings of Z\(\oplus\dots \oplus Z\)
cites work: A Remark on the Krull-Schmidt-Azumaya Theorem
cites work: Decompositions of injective modules
cites work: A Krull-Schmidt Theorem for Infinite Sums of Modules
cites work: Serial rings and finitely presented modules
cites work: Classes of modules with the exchange property
DOI: 10.1006/JABR.1996.6977
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description: scientific article; zbMATH DE number 1051515