I need to get hold of any relevant literature on the structure theory of modules over the $p$-adic integers $\mathbb{Z}_p$.
One result in particular which has been described as 'fairly standard structure theory of $\mathbb{Z}_p$-modules' is the following:
Every $\mathbb{Z}_p$-submodule of $\mathbb{Q}_p^r$ is isomorphic to $\mathbb{Q}_p^m\oplus\mathbb{Z}_p^n$ for some $m,n$.
Where might I find a proof of this result and similar ones? Do you know of a good textbook and/or any research papers covering this topic?
Start with $W_0=\Bbb{Q}_p^r$ and $V_0$ the $\Bbb{Z}_p$-submodule and $j=0$.
Take $\tilde{a}_j\in V_j-0$, let $\tilde{B}_j=\tilde{a}_j\Bbb{Q}_p\cap V_j$, then $\tilde{B}_j$ is either $\tilde{B}_j=\tilde{a}_j\Bbb{Q}_p$ or $\tilde{B}_j=\tilde{a}_j p^{-n} \Bbb{Z}_p$.
If $\tilde{B}_j=\tilde{a}_j p^{-n} \Bbb{Z}_p$ then $\tilde{a}_j p^{-n} $ is the image of some $b_j\in V_0$. Set $B_j=b_j\Bbb{Z}_p$.
If $\tilde{B}_j=\tilde{a}_j \Bbb{Q}_p$ then it is still true that there is some $b_j\in V_0$ such that $\tilde{B}_j$ is the image in $V_j$ of $B_j=b_j\Bbb{Q}_p$.
Let $W_{j+1} = W_j/\tilde{a}_j\Bbb{Q}_p, V_{j+1} = V_j/\tilde{B}_j$ (quotient group)
Note that $V_{j+1}$ is a $\Bbb{Z}_p$-submodule of $W_{j+1}$.
Replace $j$ by $j+1$ and repeat, until $V_d=0$.
Then $$V_0=\bigoplus_{j=0}^{d-1} B_j$$