Příhoda, Pavel A version of the weak Krull-Schmidt theorem for infinite direct sums of uniserial modules. (English) Zbl 1098.16004 Commun. Algebra 34, No. 4, 1479-1487 (2006). A module is called uniserial if the lattice of its submodules is a chain. In this paper, the author proves a version of the weak Krull-Schmidt theorem for infinite direct sums of uniserial modules, generalizing a corresponding result by A. Facchini and the reviewer [J. Algebra 193, No. 1, 102-121 (1997; Zbl 0885.16008)]. A uniserial module \(U\) is said to be quasi-small if whenever \(U\) is isomorphic to a direct summand of \(\bigoplus_{i\in I}M_i\), there is a finite subset \(I'\subseteq I\) such that \(U\) is isomorphic to a direct summand of \(\bigoplus_{i\in I'}M_i\). The main result of this paper states that if \(\{U_i\mid i\in I\}\) and \(\{V_j\mid j\in J\}\) are two families of nonzero uniserial modules, with \(I'=\{i\in I\mid U_i\) is quasi-small} and \(J'=\{j\in J\mid V_j\) is quasi-small}, then \(\bigoplus_{i\in I}U_i\cong\bigoplus_ {j\in J}V_j\) if and only if there is a bijection \(\sigma\colon I\to J\) and a bijection \(\tau\colon I'\to J'\) such that \(U_i\) and \(V_{\sigma(i)}\) belong to the same monogeny class for each \(i\in I\) and \(U_i\) and \(V_{\tau(i)}\) belong to the same epigeny class for each \(i\in I'\). Reviewer: Nguyen Viet Dung (Zanesville) Cited in 1 ReviewCited in 8 Documents MSC: 16D70 Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras) 16P60 Chain conditions on annihilators and summands: Goldie-type conditions Keywords:uniserial modules; weak Krull-Schmidt theorem; quasismall modules; direct summands; direct sums; monogeny classes; epigeny classes Citations:Zbl 0885.16008 PDFBibTeX XMLCite \textit{P. Příhoda}, Commun. Algebra 34, No. 4, 1479--1487 (2006; Zbl 1098.16004) Full Text: DOI References: [1] DOI: 10.1006/jabr.1996.6977 · Zbl 0885.16008 · doi:10.1006/jabr.1996.6977 [2] Facchini A., Module Theory. Endomorphism Rings and Direct Sum Decompositions in Some Classes of Modules (1998) · Zbl 0930.16001 [3] DOI: 10.1112/S0024610701002344 · Zbl 1048.16003 · doi:10.1112/S0024610701002344 [4] Příhoda P., J. Algebra This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.