Let $R$ be a principal ideal domain and let $M$ be a finetely-generated p-torsion $R$-module, i.e., every $m \in M$ satisfies $p^n m = 0$, for $p$ irreducible (equivalently prime, since $R$ is PID) and $n \ge 0$. Let $x \in M$ be an element of maximal $p$-torsion order $n$ (i.e., $p^{n}w=0$ for all $w\in M$ and $p^{n-1}x \neq 0$).
I want to show that the short exact sequence $ 0 \rightarrow Rx \rightarrow M \rightarrow M/Rx \rightarrow 0$ splits, and so $M \cong Rx \oplus M/Rx$..
I am trying to demonstrate the existence of Structure theorem for finitely generated torsion modules over a principal ideal domain, but I am not able to understand some steps of the demonstration of Mitchell, available in https://sites.math.washington.edu/~mitchell/Algf/pid.pdf page 9.
Can anyone help me understand the demonstration or show me another way to do it.
Thank you in advance.
Since $\tilde{L}\supsetneq L$, we automatically have $l(\tilde{L})>l(L)$ by definition.
For the other point, take any $z\in\tilde{L}-L$, then there is $n>0$ such that $p^nz = 0\in L$, thus we can take $$k := \min\{m\in\mathbb{Z}\mid p^kz\in L\}\ge1$$ and set $y:= p^{k-1}z$, and we're done.