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I’m trying to show that a finitely generated flat module over a PID is torsion-free. I know the converse is a consequence of the structure theorem for finitely generated modules over a PID and the fact that free modules are flat, but I’m not sure if that theorem is useful here. Any hints would be appreciated.

dahemar
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    Over any integral domain $R$, flat modules are torsion-free. This just follows from tensoring the multiplication map $R \xrightarrow{\cdot a} R$ with a flat module $M$ and using the isomorphism $R \otimes_R M \cong M$. But I'm sure this has been answered before. The structure theorem is overkill – Lukas Heger Dec 10 '22 at 17:41
  • @LukasHeger Thank you! I had looked for similar questions prior to asking but didn’t find that one. Yes, it does. – dahemar Dec 10 '22 at 18:02
  • @LukasHeger The idea is that if the multiplication map is injective then the map obtained by tensoring would be injective too, right? – dahemar Dec 10 '22 at 18:06
  • exactly, that's the idea – Lukas Heger Dec 10 '22 at 18:08

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