Subrings of $\mathbb{Z}[t]\mathrm{/}(t^4)$

In this note we study the distribution of the subrings of $\mathbb Z[t]/(t^4)$ and prove two results. The first result gives an asymptotic formula for the number of subrings of $\mathbb{Z}[t]\mathrm{/}(t^4)$ of bounded index. The method of proof of this theorem is $p$-adic integration a la Grunewald, Segal, and Smith. Our second result is about the distribution of cocyclic subrings in $\mathbb{Z}[t]\mathrm{/}(t^4)$. Our proof of this result is combinatorial and is based on counting certain classes of matrices with Smith normal forms of a special form.