  We count subrings of small index of $\mathbb{Z}^n$, where the addition and
multiplication are defined componentwise. Let $f_n(k)$ denote the number of
subrings of index $k$. For any $n$, we give a formula for this quantity for all
integers $k$ that are not divisible by a 9th power of a prime, extending a
result of Liu.
