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Idempotents of $\mathbb{Z}_n$

Suman Mondal, Dhiren Kumar Basnet

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 $n \in \mathbb{N}$.

Idempotents of $\mathbb{Z}_n$

Abstract

We know that if there are distinct prime factors of , then the ring of integers modulo has exactly idempotent elements. In this article, we try to describe all the idempotents of for any given .

Paper Structure

This paper contains 5 sections, 15 theorems, 4 equations.

Table of Contents

  1. Introduction
  2. preliminaries
  3. The Main Result
  4. Some Further Investigation
  5. Final Results

Key Result

Theorem 2.1

Let $n=2\cdot m$, where $m(\geq 3) \in \mathbb{Z}$ and $(m,2)=1$, then $\overline{m}$ and $\overline{m+1}$ are nontrivial idempotents of $\mathbb{Z}_{n}$.

Theorems & Definitions (34)

  • Theorem 2.1
  • proof
  • Corollary 2.1.1
  • proof
  • Example 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Example 2.2
  • Example 2.3
  • Example 2.4
  • ...and 24 more