---
date: 2022-01-23 19:52
modification date: Sunday 23rd January 2022 19:53:25
title: fractional ideal
aliases: ["fractional ideal", "colon ideal", "principal fractional ideal"]

---

Tags: 
#todo #todo/stub 
Refs:
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# Fractional Ideal
Idea: get methods that work for primes $p\in \ZZ$ to work for ideals $\gens{p} \in \Id(\ZZ)$, and then generalize to other rings e.g. to get unique factorization.
![](attachments/Pasted%20image%2020220123202618.png)
![](attachments/Pasted%20image%2020220123202646.png)
Definitions:
![](attachments/Pasted%20image%2020220123195231.png)
![](attachments/Pasted%20image%2020220123195314.png)

# Colon Ideal
![](attachments/Pasted%20image%2020220123195354.png)

# Invertibility

Invertibility of fractional ideals:
![](attachments/Pasted%20image%2020220123195729.png)

Fact: in a [Dedekind domain](Dedekind%20domain.md) every nonzero fractional ideal is invertible.

Example of noninvertible ideals:
![](attachments/Pasted%20image%2020220123195920.png)
![](attachments/Pasted%20image%2020220123201959.png)

# Primality

![](attachments/Pasted%20image%2020220124114019.png)
# Exercises

Let $A$ be an integral domain with fraction field $K$ and let $M$ be a nonzero A-submodule of $K$. Then $M^{\vee} \simeq(A: M):=\{x \in K: x M \subseteq A\}$; in particular, if $M$ is an invertible fractional ideal then $M^{\vee} \simeq M^{-1}$ and $M^{\vee \vee} \simeq M$.