Let R be a commutative ring with 1. If there exists a non-zero R-module M such that every submodule of M is free, then R is a PID.(adsbygoogle = window.adsbygoogle || []).push({});

I remember proving something similar to this, assuming submodules of all R-modules are free, but I'm not too sure about this question. The direction I am headed in is to consider M as an I-module. As IM->N, N has a basis {n_i}. After playing around a bit I get lost.

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# Nonzero R-Module over commuttaive ring, all submodules free => R PID?

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