---
tag: E190TAV2S6
kind: concept
name: Invertible ideal
msc: [13A15, 13F05, 13C20]
sources:
  - url: https://en.wikipedia.org/wiki/Fractional_ideal
---

## Summary

An invertible ideal of an [[integral domain|0QVD7WBFGN]] $R$ is a fractional ideal $I$ that has an inverse $J$ with $IJ = R$.
Fractional ideals are $R$-submodules of the [[field of fractions|0F2PTGX1A2]] that become ordinary [[ideals|096Z9MP2ME]] after clearing one denominator.
The invertible fractional ideals form a group under multiplication, and its quotient by the principal ones is the [[Picard group|DZW58R6T2Q]] of $R$.
A [[Dedekind domain|7NPY2CN5KB]] is exactly a domain in which every nonzero fractional ideal is invertible.

## Explanations

### L1 human

Ideals can be multiplied, but in general they cannot be divided.
An invertible ideal is one that can be divided by: some fractional ideal multiplies it back to the whole ring.
Every nonzero principal ideal $(a)$ is invertible, with inverse $(1/a)$.
In $\Z[\sqrt{-5}]$, the ideal $P = (2, 1 + \sqrt{-5})$ is not principal, but $P^2 = (2)$, so $\frac{1}{2} P$ is an inverse of $P$.