A duplicate of this:
Let $R$ be a finite local principal ideal ring that is not a field (I'm going to use $\mathbb{Z}/p^n\mathbb{Z}$ as an example, in the general case the only proper ideals are the non-zero powers of the maximal ideal) and $M$ a finitely generated free $R$-module. Let $S_i$ be the set the submodules of $(p^i)M$, for $0 \leq i \leq n-1$.
Is it true that $S_i$ contains only the submodules in $S_{i-1}$ and their "stretched" versions, that is the submodules in $S_{i-1}$ in which we also allow coefficients from $(p^{i-1})$ instead of just $p^i$? I was thinking that, in the end, we can completely determine all submodules of M by just looking at $S_{n-1}$ and "stretching", is that true? I'm thinking the answer should be yes (for example, using the Smith normal form, or even a direct counting argument), but I may be overlooking something. Thanks!
