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

Benjamin Krakoff; Julia Menzel; Nathan Kaplan; Stanislav Atanasov

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

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arxiv 1609.06433 v5 pith:MPCRVP74 submitted 2016-09-21 math.CO math.NT

Stanislav Atanasov , Nathan Kaplan , Benjamin Krakoff , Julia Menzel This is my paper · ORCID

classification math.COmath.NT

keywords indexsubringsmathbbadditioncomponentwisecountcountingdefined

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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.

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