I am currently working on the following proof from Milne's Class Field Theory.
Here $K$ is a nonarchimedean local field, $A$ its valuation ring $A = \{\alpha \in K: | \alpha| \leq 1\}$. So $A$ is a DVR and its maximal ideal $\mathcal{m} = \{ \alpha \in A: | \alpha| < 1\}$ is generated by some prime element $\pi$. $A/ (\pi)$ is a finite field with $q$ elements.
The highlighted text is the only part that is not clear to me. I can see that $M_n/M_1 \simeq M_{n-1}$ by exactness, and by induction we may even assume that $M_1$ and $M_{n-1}$ are cyclic. $M_n$ decomposes into a direct sum of cyclic modules (since $A$ is a PID with a unique prime element), but why is it necessary that $M_n$ is cyclic? I feel like I am missing something very obvious and I would appreciate any help!