---
tag: YB4JTB4WJN
kind: concept
name: Fractional ideal
msc: [13A15, 11R04]
sources:
  - url: https://en.wikipedia.org/wiki/Fractional_ideal
same_as:
  wikidata: Q1497184
  wikipedia: Fractional ideal
---

## Summary

A fractional ideal of an integral domain $R$ is a subset of its field of fractions.
After clearing denominators, it becomes an [[ideal|096Z9MP2ME]] of $R$.
Fractional ideals can be multiplied, and in a [[Dedekind domain|7NPY2CN5KB]] every nonzero fractional ideal has an inverse.
So the nonzero fractional ideals of a Dedekind domain form a group, and its quotient by the principal ones is the [[ideal class group|WQ46BD3ARY]].

## Explanations

### L1 human

Ideals of the integers are the sets $n\Z$ of multiples of a number $n$.
They can be multiplied, $(2\Z)(3\Z) = 6\Z$, but not divided: there is no ideal $I$ of $\Z$ with $(2\Z) I = \Z$.
Fractional ideals allow fractions: $\tfrac12 \Z = \set{\ldots, -\tfrac12, 0, \tfrac12, 1, \ldots}$ is one, and $(2\Z)(\tfrac12\Z) = \Z$.
So fractional ideals are to ideals what fractions are to integers.