If $R$ is a ring and, further, a PID, a morphism of $f : M \to N$ of finitely generated, free $R$-modules has a Smith normal form. Does this also hold when $M$ and $N$ are finitely generated but not free? For some reason I have found it difficult to find a reference of this.
The main example I'm interested in is from TDA where when we work over a polynomial ring $\mathbb{k}[x]$ with integer exponents. In particular, persistent chain complexes can be thought of as chain complexes $Ch(\mathbb{k}[x])$ over the graded ring $\mathbb{k}[x]$. The boundary operator $d_k : C_k \to C_{k-1}$ is a graded morphism of $\mathbb{k}[x]$-modules, and the Smith normal of the matrix form determines a bijective pairing between bases for the coimage and image of $d_k$ which corresponds to the persistence pairing.
I'm wondering in particular if this still works when $C_k$ and $C_{k-1}$ are finitely generated but not free. In the original Carlsson and Zomorodian paper, the chain complexes arise from filtered complexes and are thus always free, so the case I'm interested in is never discussed.