Let $f \in \Bbb Z[x]$ be a monic polynomial, $n$ is a positive integer. Consider all finitely generated $O=\Bbb Z[x]/(f(x))$ module $M$ such that $M \cong \Bbb Z^n$ as $\Bbb Z$-module, are there only finitely many isomorphism classes of such $M$ ?
If $f$ is irreducible then $O$ is an order in a number field. Furthermore, assume $O$ is the algebraic integer ring in this number field then the answer is positive by finiteness of class number.
I tried to deal with the general case, but there are several problems such as that a torsion free module may not be flat and that $\Bbb Z[x]/(x^2-1) \not \cong \Bbb Z[x]/(x-1) \oplus \Bbb Z[x]/(x+1)$). Is there any counterexample? If not, how do we compute the number of isomorphism classes using class number?
Edit: Thanks for the answer below, there exists some counterexamples.
Here is some of my ideas about a possible positive solution for some cases after posting this question.
Firstly, as $\Bbb Z[x]$ is a $2$-dimensional regular ring, by standard argument (classification of f.g modules over Iwasawa algebra up to pseudo-isomorphism) there exists a morphism from $M$ to $\bigoplus_{i} \Bbb Z[x]/f_i^{n_i}$ which has finite kernel and cokernel and $f_i$ is irreducible. One example is the standard map from $\Bbb Z[x]/(x^2-1)$ to $\Bbb Z[x]/(x-1) \oplus \Bbb Z[x]/(x+1)$ with cokernel $\Bbb Z/2$. Maybe good controls for kernel and cokernel can help.
A similiar idea is to regard $M$ as a coherent sheaf over $X=\text{Spec} \ O$, so when $O$ is an order , there exists an open subset $U$ of $X$ such that $X_U$ is normal with finite picard group and we have good understanding for $M|_U$. As there also exists a structure theorem for f.g modules on $PIR$ (Pricinpal Ideal Ring), this can also be done for general $O$. The remaining problem is to analysis $M$ on $X-U$ (a finite set).
This is not true. For instance, let $f(x)=x^2$, let $n$ be a positive integer, and consider the module $O_n=\mathbb{Z}^2$ with $x$ acting by the matrix $\begin{pmatrix} 0 & n \\ 0 & 0\end{pmatrix}$. These modules are all non-isomorphic because the torsion subgroup of $O_n/xO_n$ has $n$ elements.