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

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.

29 pages. To appear in Acta Arithmetica

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Cited by in corpus (1)

• On solutions of ${\sum }_{i=1}^{n}1/x_i=1$ in integers of the form $2^a{k}^b$