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Counting finite index subrings of $\mathbb Z^n$

Volume 197 / 2021

Stanislav Atanasov, Nathan Kaplan, Benjamin Krakoff, Julia H. Menzel Acta Arithmetica 197 (2021), 221-246 MSC: Primary 20E07; Secondary 11H06, 11M41. DOI: 10.4064/aa180201-29-7 Published online: 9 November 2020


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.


  • Stanislav AtanasovDepartment of Mathematics
    Columbia University
    Room 408, MC4406
    2990 Broadway
    New York, NY 10027, U.S.A.
  • Nathan KaplanDepartment of Mathematics
    University of California, Irvine
    340 Rowland Hall
    Irvine, CA 92697, U.S.A.
  • Benjamin KrakoffDepartment of Mathematics
    University of Michigan
    East Hall B723
    530 Church Street
    Ann Arbor, MI 48109, U.S.A.
  • Julia H. MenzelProgram in History, Anthropology,
    and Science, Technology, and Society
    Room E51-163
    77 Massachusetts Avenue
    Cambridge, MA 02139, U.S.A.

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