Counting finite index subrings of $\mathbb Z^n$
Volume 197 / 2021
Acta Arithmetica 197 (2021), 221-246
MSC: Primary 20E07; Secondary 11H06, 11M41.
DOI: 10.4064/aa180201-29-7
Published online: 9 November 2020
Abstract
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.
