---
tag: 437YRCNYRJ
kind: concept
name: Radical of an ideal
msc: [13A15]
sources:
  - url: https://en.wikipedia.org/wiki/Radical_of_an_ideal
same_as:
  wikidata: Q1199022
  wikipedia: Radical of an ideal
---

## Summary

The radical of an [[ideal|0MYSA6MYNA]] $I$ of a commutative ring is the set of elements some power of which lies in $I$.
It is again an ideal, and it contains $I$.
The radical is the intersection of the [[prime ideals|06HC6HK0Z8]] that contain $I$.
It is central in the link between ideals and geometry given by [[Hilbert's Nullstellensatz|D85VZWS9QK]].

## Explanations

### L1 human

The radical forgets multiplicities.
In $\Z$, the ideal $12\Z$ records the number $12 = 2^2 \cdot 3$, and its radical is $6\Z$, which only records the primes $2$ and $3$.
An integer $r$ has a power divisible by $12$ exactly when $r$ is divisible by $2$ and by $3$.
For polynomials, the ideals $(x^2)$ and $(x)$ have the same zero set, and the radical of $(x^2)$ is $(x)$.