I have been reading the book Algebraic Number Theory by J.W.S. Cassels & A. Frohlich and I am currently stuck in a proof about Localization of Modules over Dedekind domains.
Let $R$ be a Dedekind domain, $p$ one of its prime ideal, and $U=frac(R)$ its quotient field. Let $V$ be a vector space over $U$ and $M$ a $R_p$-submodule such that $M$ spans $V$ (it means $M$ contains a basis for $V$).
Then I know that I can use modules over PID to decompose $M=\left(\bigoplus_{i=1}^nR_pv_i\right)\oplus \left(\bigoplus_{\text{some }k}R_p/(p^k)\right)$. Then because of dimension counting, we conclude that the torsion term should not exist or else the dimension of the module will be greated than that of $V$, which should not exist in modules over PID.
Is my proof correct or not? I have seen a similar question (link) asked before for examples of submodules not free in Dedekind domains. So I guess it means that if the underlying ring $R_p$ is a PID, then all submodules of finite-dimension vector spaces should be free?
Thank you very much for your time.