I'm searching for a proof of the following statement
Let $M$ be a $P$-torsion module over a Dedekind domain $R$, and let $x_1 \in M$ such that $\operatorname{Ann}_R(x_1) = Ann_R(M)$. If there exist $y_2 \dots y_s \in M/Rx_1$ such that $$ \frac{M}{Rx_1} \simeq Ry_2 \oplus \cdots \oplus Ry_s $$ then there exist $x_2, \dots x_s \in M$ such that $\operatorname{Ann}_R(x_i) = \operatorname{Ann}_R(y_i)$, $\pi(x_i) = y_i$ with $\pi: M \to M/Rx_1$ the projection, and $$ M \simeq Rx_1 \oplus \cdots \oplus Rx_s $$
I'm somewhat familiar with the lemma for principal domains, in which for a given $y \in M/Rx_1$ we first show that it can be chosen $z \in M$ with $\operatorname{Ann}_R(z) = \operatorname{Ann}_R(y)$ and $\pi(z) = y$, from which the ulterior condition follows.
However, the aforementioned construction uses strongly that if $\operatorname{Ann}_R(M) = p^mR$, then $p^m$ annihilates $M/Rx_1$ and so $p^my = ax_1$ for some $a \in R$.
My knowledge on Dedekind domains is very rudimentary so I would really appreciate if you could either provide a proof of this statement or a reference in which I could find one. Maybe the same be done for these type of rings? That is, if $\operatorname{Ann}_R(M) = \mathcal{P}^m$, can we find $p \in \mathcal{P}$ with $p^my = ax_1$? I think I could take it from there following the proof for PIDs.
$\newcommand\ideal[1]{\mathfrak{#1}}\DeclareMathOperator\Ann{Ann}$Let $R$ be a Dedekind domain, $\ideal p$ be a prime ideal in $R$ and $M$ be a $\ideal p$-torsion $R$-module and $x_1\in M$ such that $\Ann_R(x_1)=\Ann_R(M)$.
Then $\Ann_R(M)\supseteq\ideal p^n$ for some $n\in\Bbb N$, hence $M$ can regarded as a $R/\ideal p^n$-module.
Let $R_{\ideal p}$ denote the localization at $\ideal p$ of $R$. Then $R_{\ideal p}$ is a PID, $\ideal m=R_{\ideal p}\ideal p$ is a maximal ideal and $R_{\ideal p}/\ideal m^n\cong R/\ideal p^n$. Consequently, $M$ is a $\ideal m$-torsion $R_{\ideal p}$-module and hence the proof valid for PIDs applyies.