The Krull-Schmidt property for ideals and modules over integral domains.
type: Item
instance of: scholarly article
title: The Krull-Schmidt property for ideals and modules over integral domains.
zbMATH Open document ID: 1035.13005
author: Pat Goeters
author: Bruce Olberding
published in: Rocky Mountain Journal of Mathematics
publication date: 2003-12-03T00:00:00Z
full work available at URL: CONT32-4.html
review text: Let \(R\) be a commutative integral domain and \(\mathcal{C}\) a class of \(R\)-modules. The Krull-Schmidt property holds for \(\mathcal{C}\) if, whenever \[ G_1\oplus G_2\oplus\cdots\oplus G_n\cong H_1\oplus H_2\oplus\cdots\oplus H_m \] for \(G_i, H_j\in\mathcal {C}\), then \(n=m\) and, after reindexing, \(G_i\cong H_i\) for all \(i\leq n\). If, instead of \(G_i\cong H_i\), we require only that \(k>0\) exists such that \(G_i^{(k)}\cong H_i^{(k)}\) for all \(i\), then we say the weak Krull-Schmidt property holds for \(\mathcal{C}\) (we write \(G^{(k)}\) for a direct sum of \(k\) copies of a module \(G\).) In the paper under review the authors examine when versions of the Krull-Schmidt property hold for the following: (1) direct sums of ideals of integral domains, (2) direct sums of indecomposable submodules of finitely generated free modules, (3) direct sums of rank one torsion-free modules.
reviewed by: Siamak Yassemi
Mathematics Subject Classification ID: 13C05
zbMATH DE Number: 2012081
zbMATH Keywords: Krull-Schmidt property
zbMATH Keywords: integral domain
MaRDI profile type: Publication
OpenAlex ID: W2091627937
cites work: Almost Maximal Integral Domains and Finitely Generated Modules
cites work: Module equivalences: Local to global. When primitive polynomials represent units
cites work: Q4513600
cites work: Overrings of Prüfer domains. II
cites work: Unique decompositions into ideals for noetherian domains.
cites work: Krull-Schmidt theorems in dimension 1
cites work: Package deal theorems and splitting orders in dimension 1
cites work: Q4052239
cites work: Globalizing local properties of Prüfer domains
cites work: Prüfer domains and pure submodules of direct sums of ideals
cites work: A geometric setting for some properties of torsion-free modules.
cites work: Rings with two-generated ideals
cites work: Multiply Maximally Complete Fields
DOI: 10.1216/RMJM/1181070031
description: scientific article; zbMATH DE number 2012081