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For finitely generated modules over a PID, there is the structure theorem (https://en.wikipedia.org/wiki/Structure_theorem_for_finitely_generated_modules_over_a_principal_ideal_domain) that allows us to write the module as a direct sum of its free part and torsion part.

Is there any further generalisation that is more general than finitely generated modules over a PID?

Also another related question: Is there a necessary and sufficient condition that allows us to write a module as a direct sum of its free and torsion submodules?

Thanks.

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This doesn't quite answer your question, but I think it is close enough to be relevant.

The following is exercise 19.6(b) in Eisenbud:

Let $R$ be a Dedekind domain and $M$ be a finitely generated $R$-module, then $M \cong M_{tors} \oplus M/M_{tors}$. Furthermore $M/M_{tors}$ is projective (but not necessarily free). As projective modules over PIDs are free, this reduces to the proposition you mention.

There is also a structure theorem for finitely generated modules over a Dedekind domain, very similiar to the one for PIDs.

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