  In this paper we focus on modules over a finite chain ring $\mathcal{R}$ of
size $q^s$. We compute the density of free modules of $\mathcal{R}^n$, where we
separately treat the asymptotics in $n,q$ and $s$. In particular, we focus on
two cases: one where we fix the length of the module and one where we fix the
rank of the module. In both cases, the density results can be bounded by the
Andrews-Gordon identities. We also study the asymptotic behaviour of modules
generated by random matrices over $\mathcal{R}$. Since linear codes over
$\mathcal{R}$ are submodules of $\mathcal{R}^n$ we get direct implications for
coding theory. For example, we show that random codes achieve the
Gilbert-Varshamov bound with high probability.
