Let $R$ be a complete local domain of dimension $1$. Let $M$ be a finitely generated torsion-free indecomposable $R$-module. Then, must $M$ be isomorphic to an ideal of $R$?
Also clearly, $R$ embeds in End$_R(M)$.
2026-03-25 07:44:03.1774424643
Finitely generated torsion-free indecomposable modules over one-dimensional complete local domains are isomorphic to ideal?
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It is known that a reduced local ring of dimension 1 has bounded Cohen-Macaulay type if and only if it has finite Cohen-Macaulay type. Any such domain for which indecomposable torsion-free modules are ideals must have bounded CM type since the multiplicity of any (nonzero) ideal is that of $R$. So any CM local ring of dimension 1 that does not have finite CM representation type (there is a classification showing only a few of finite type exist) gives an example. See here as a starting point for references. https://arxiv.org/pdf/math/0211411.pdf