---
tag: NZ1G9WRWFB
kind: concept
name: Nil ideal
msc: [16N40]
sources:
  - url: https://en.wikipedia.org/wiki/Nil_ideal
same_as:
  wikidata: Q453275
  wikipedia: Nil ideal
---

## Summary

A nil ideal of a [[ring|0B1261B5ED]] is a one-sided or two-sided ideal all of whose elements are [[nilpotent|0ZB2JG1M9C]].
Nil ideals are a main object of radical theory, the study of the parts of a ring that are discarded to reach simpler rings.
In a commutative ring the largest nil ideal is the [[nilradical|QQDN6EYQNJ]].
Whether every nil one-sided ideal lies in a nil two-sided ideal is the open [[Köthe conjecture|RT2D9JEVRR]].

## Explanations

### L1 human

An element is nilpotent when some power of it is zero.
A nil ideal is an ideal in which every element has this property, though the power needed may depend on the element.
In the ring $\Z/8\Z$, the ideal of even residues is nil: every even residue $a$ satisfies $a^3 = 0$.
In $\Z$, the ideal $2\Z$ is not nil, since no power of $2$ is zero.