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Basic Ring Theory - 22 (Flatness and Torsion)

Posted: Sun May 22, 2016 1:23 am

by Tsakanikas Nickos

Let $A$ be a ring and let $M$ be an $A$-module. Show the following:

If $A$ is an integral domain and $M$ is flat over $A$, then $M$ is torsion-free.
If $A$ is a P.I.D., then $M$ is flat over $A$ if and only if $M$ is torsion-free.
If $A$ is a Dedekind domain, then $M$ is flat over $A$ if and only if $M$ is torsion-free.

Comment: Hence, over P.I.D.'s and Dedekind domains, flatness has a nice characterization.