---
tag: QQDN6EYQNJ
kind: concept
name: Nilradical
msc: [13A15]
sources:
  - url: https://en.wikipedia.org/wiki/Nilradical_of_a_ring
same_as:
  wikidata: Q3341881
  wikipedia: Nilradical of a ring
---

## Summary

The nilradical of a commutative ring is the set of its [[nilpotent elements|0N9R5NN6TW]].
It is an ideal, the [[radical|437YRCNYRJ]] of the zero ideal.
It equals the [[intersection of all prime ideals|3CZ8PXR9JY]] of the ring, and a ring with zero nilradical is called [[reduced|SZHY975FFX]].

## Explanations

### L1 human

An element is nilpotent when some power of it is zero.
The nilradical collects all such elements.
In $\Z/12\Z$, the nilpotent elements are $0$ and $6$, since $6^2 = 36 = 0$ in $\Z/12\Z$, while for instance no power of $2$ or $4$ is divisible by $3$.
In $\Z$ or in any field the nilradical is $\set{0}$.
Geometrically, nilpotent functions vanish at every point, so the nilradical is invisible to the points of the [[spectrum|FB45BHHA2J]].