Counting Finite Index Subrings of $\mathbb{Z}^n$

TL;DR

This paper provides a formula for counting subrings of small index in ^n, extending previous results to a broader class of integers, thereby advancing understanding of algebraic substructure enumeration.

Contribution

It introduces a general formula for counting subrings of ^n of a given index, applicable to all integers not divisible by a ninth power of a prime, extending Liu's earlier work.

Findings

Derived a formula for subring counts for most integers

Extended Liu's results to a wider class of indices

Enhanced understanding of subring enumeration in ^n

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.