This is a follow-up to my previous question, only this time about a finitely generated torsion module $T$ over a Dedekind domain $R$.
What I have done is looked at the prime factorization of $ann(T)\le R$, which looks like $ann(T)=\prod_iP_i^{k_i}$. I then used the Chinese Remainder Theorem to write $T\cong\oplus_iT^{P_i}$, where
$$ T^{P_i}=\{t\in T\mid P_i^{k_t}t=0\text{ for some }k_t\}$$
That is, $T^{P}$ is all the elements in $T$ annihilated by some power of $P$. I then showed that $T\otimes R_{P}\cong T^{P}$ as $R$-modules. I also showed there was an isomorphism as $R_{P}$-modules
$$ T\otimes R_P\cong \oplus_{i=1}^n R_P/P^{e_i} $$ where $e_{i+1}\ge e_i$.
And finally that, as $R$-modules, $R_P/P^{e_i}\cong R/P^{e_i}$.
So to show uniqueness, I have been using the following strategy: if $f(j)=\#(e_i=j)$, then I want to show the function $f$ is an invariant of $T^P$ that doesn't depend on any decomposition. However, all my attempts have failed. I know $R_P$ is a PID, so I could use the general structure theorem here. But $R_P$ is a DVR, with only one prime, so it seems like this case should be easier, and doable directly.
Does anyone have any suggestions about how to proceed?