---
tag: S4GQN5V54X
kind: concept
name: Radical ideal
msc: [13A15]
sources:
  - url: https://en.wikipedia.org/wiki/Radical_of_an_ideal
---

## Summary

A radical ideal of a commutative ring is an [[ideal|0MYSA6MYNA]] that equals its own [[radical|437YRCNYRJ]].
Equivalently, if some power of an element lies in the ideal, then the element itself lies in it.
Every [[prime ideal|06HC6HK0Z8]] is radical.
By [[Hilbert's Nullstellensatz|D85VZWS9QK]], the radical ideals of a polynomial ring over an algebraically closed field correspond to the algebraic sets.

## Explanations

### L1 human

A radical ideal has no "repeated factors".
In $\Z$, the ideal $n\Z$ with $n \ge 1$ is radical exactly when $n$ is square-free, such as $6$ or $30$.
The ideal $12\Z$ is not radical: $6^2 = 36$ lies in it, but $6$ does not.
The quotient by a radical ideal is a [[reduced ring|SZHY975FFX]], a ring without nonzero nilpotent elements.