Idempotents of $\mathbb{Z}_n$

TL;DR

This paper explores the structure of idempotent elements in the ring of integers modulo n, providing a comprehensive description for any natural number n.

Contribution

It extends the known result about the number of idempotents to a detailed characterization of all idempotents in al_n for any n.

Findings

Number of idempotents equals 2^k for n with k prime factors

Provides explicit description of all idempotents in al_n

Generalizes previous results to all natural numbers n

Abstract

We know that if there are $k$ distinct prime factors of $n \in \mathbb{N}$, then the ring $\mathbb{Z}_n$ of integers modulo $n$ has exactly $2^k$ idempotent elements. In this article, we try to describe all the idempotents of $\mathbb{Z}_n$ for any given $n \in \mathbb{N}$.