Let $R$ be an $I$-adically separated and complete valuation ring, with $I$ finitely generated.


It is used a few times in Bosch, [Lectures on Formal and Rigid Geometry](https://www.math.purdue.edu/%7Etongliu/seminar/rigid/Bosch.pdf) e.g. first lines of pg. 164, Cor. 5 and Cor. 6 (their condition (V) is what I stated above) that



> 
> If an $A$ module has no $I$ torsion then it is flat over $R$.
> 
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> 


I don't see why this is true. Any suggestions / references would be appreciated.




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What I thought: We know $A$ is flat over domain $R$ iff it is $R$-torsion free.


If the statement were true: $I$-torsion free $\Rightarrow$$R$-torsion free.


This would hold if $I$ is a maximal ideal but otherwise I don't see why.



Flatness criterion for $I$-adic ring: $I$-torsion free
In Section 7.3 it is assumed that $I$ is the ideal of definition or $R$. It follows that the $I$-adic topology is separated, so (because $R$ is a valuation ring) every nonzero ideal of $R$ contains some power of $I$, so everything that is $R$-torsion is $I$-torsion.


