In graduate school, while I was working on the maximal subgroup growth of certain metabelian groups, I discovered and proved a lemma which gave me the impression that it was already known. Do you know of any reference for this result? Does it follow easily from a known result? Here is the result:

Let $N$ be a finitely generated $\mathbb{Z}[x]$-module. Suppose (by using the fundamental theorem of f.g. modules over PIDs) we have the following isomorphism of $\mathbb{Q}[x]$-modules:

$$\mathbb{Q} \otimes_\mathbb{Z} N \cong \left( \bigoplus_{j=1}^{s_1} \mathbb{Q}[x]/(a_j) \right) \oplus \mathbb{Q}[x]^{s_2} $$ for some $a_j \in \mathbb{Q}[x]$ that are not units and such that $a_1 \mid a_2 \mid \ldots \mid a_{s_1}$.

Then for all large primes $p$, we have the following isomorphism of $\mathbb{F}_p[x]$-modules:

$$N/pN \cong \left( \bigoplus_{j=1}^{s_1} \mathbb{F}_p[x]/(\overline{a_j}) \right) \oplus \mathbb{F}_p[x]^{s_2}.$$

While my proof of the above result is about three pages long, I do have a short heuristic argument:

When doing the computation required in finding the decomposition of $\mathbb{Q} \otimes N$ into a direct sum of cyclic modules, the only thing keeping us from doing this computation to $N$ itself (as a $\mathbb{Z}[x]$-module) is that we may need to divide by finitely many integers.

So if $p$ is large enough, then in $\mathbb{F}_p$ we can divide by all those integers (i.e. their residues mod $p$). For such $p$, the steps of the algorithm would be the same for $N/pN$ as for $\mathbb{Q} \otimes N$.

In case you are interested, this is Lemma 41 in my arXiv paper Maximal subgroup growth of some metabelian groups. I actually proved a slight generalization, concerning finitely generated $D^{-1}\mathbb{Z}[x]$-modules, where $D$ is the multiplicative closure of a finite set of primes.