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A version of the weak Krull-Schmidt theorem for infinite direct sums of uniserial modules. (English) Zbl 1098.16004

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 \(i\in I'\).

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

Citations:

Zbl 0885.16008
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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
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