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Let $R$ be a commutative principal ideal ring (not necessarily Artinian) and let $M$ be a finitely generated $R$-module. Is $M$ a direct sum of cyclic $R$-modules? (i.e. a generalization of the theory of f.g. modules over a PID). I read somewhere that this is true, but I cannot find a reference where this is proven.

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    $\begingroup$ See the answer by Luc Guyot to the MO question mathoverflow.net/questions/22722. $\endgroup$
    – KConrad
    Jan 4, 2022 at 6:44
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    $\begingroup$ Direct link to this very answer mathoverflow.net/a/243540/14094 $\endgroup$
    – YCor
    Jan 4, 2022 at 11:05
  • $\begingroup$ Thank you very much! $\endgroup$ Jan 4, 2022 at 13:10
  • $\begingroup$ That answer is what I am looking for. I did a search after seeing it and also found the result spelled-out in theorem 15.33 in the book "Matrices over Commutative Rings" by William Brown $\endgroup$ Jan 4, 2022 at 13:17

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