Counting finite index subrings of $\mathbb Z^n$
نویسندگان
چکیده
We count subrings of small index $\mathbb {Z}^n$, where the addition and multiplication are defined componentwise. Let $f_n(k)$ denote number $k$. For any $n$, we give a formula for this quantity all integers $k$ that
منابع مشابه
Counting Subrings of Z of index k for small n
In this article we investigate subring growth of Z for small n using p-adic integration. To the memory of Fritz Grunewald
متن کاملCounting Finite Index Subgroups
Let F be a finitely generated group. Denote by an(T) (resp.<rn(r)) the number of subgroups of F of index n (resp. of index at most n). This paper deals with the connection between the algebraic structure of the group F and the arithmetic properties of the sequence an(F), n — 1,2,3,..., e.g., the growth of the sequence an(T) ("the subgroup growth") or the properties of the function Cr(s) = X2£Li...
متن کاملCounting Roots of Polynomials over $\mathbb{Z}/p^2\mathbb{Z}$
Until recently, the only known method of finding the roots of polynomials over prime power rings, other than fields, was brute force. One reason for this is the lack of a division algorithm, obstructing the use of greatest common divisors. Fix a prime p ∈ Z and f ∈ (Z/pZ)[x] any nonzero polynomial of degree d whose coefficients are not all divisible by p. For the case n = 2, we prove a new effi...
متن کاملon supersolvability of finite groups with $mathbb p$-subnormal subgroups
in this paper we find systems of subgroups of a finite group, which $bbb p$nobreakdash-hspace{0pt}subnormality guarantees supersolvability of the whole group.
متن کاملOn Zero Subrings and Periodic Subrings
We give new proofs of two theorems on rings in which every zero subring is finite; and we apply these theorems to obtain a necessary and sufficient condition for an infinite ring with periodic additive group to have an infinite periodic subring. 2000 Mathematics Subject Classification. 16N40, 16N60, 16P99. Let R be a ring and N its set of nilpotent elements; and call R reduced if N = {0}. Follo...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Acta Arithmetica
سال: 2021
ISSN: ['0065-1036', '1730-6264']
DOI: https://doi.org/10.4064/aa180201-29-7