# Contents

## Idea

A square-free number is a natural number $n$ such that for all prime numbers $p$, $p^2$ does not divide $n$.

## Properties

Given a square-free number $n$ with prime factors

$n = \prod p_i$

the completion of the sequence of finite cyclic rings $\mathbb{Z}/n^i\mathbb{Z}$ is the ring $\mathbb{Z}_n$, which is a subring of the profinite integers and the product of the p-adic integers of the prime factors of $n$:

$\mathbb{Z}_n \coloneqq \prod \mathbb{Z}_{p_i}$

Given a square-free integer $n$, the integers modulo n $\mathbb{Z}/n\mathbb{Z}$ is a reduced ring. Since for every integer $n$ the $\mathbb{Z}/n\mathbb{Z}$ is a prefield ring, $\mathbb{Z}/n\mathbb{Z}$ is an integral domain and thus a field if and only if $n$ is a prime number.

Given any odd prime number $p$, the integer $2 p$ is square-free, and the ring $\mathbb{Z}/2 p\mathbb{Z}$ has idempotent elements of $0$, $1$, $p$, and $p + 1$. Every element of $\mathbb{Z}/2 p\mathbb{Z}$ could be written as a linear combination of $p$ and $p + 1$.