---
tag: 3G2JQEYBEK
kind: concept
name: Upper nilradical
msc: [16N40]
sources:
  - url: https://en.wikipedia.org/wiki/Nilradical_of_a_ring
---

## Summary

The upper nilradical of a ring is its largest [[nil ideal|NZ1G9WRWFB]], the sum of all two-sided ideals that consist of nilpotent elements.
It contains the [[prime radical|WF117EJ2HF]] and the [[Levitzki radical|9DD8M9R868]].
For a [[commutative ring|0310REJJY8]] all three coincide with the [[nilradical|QQDN6EYQNJ]].
Whether the upper nilradical also contains every nil one-sided ideal is the open [[Köthe conjecture|RT2D9JEVRR]].

## Explanations

### L1 human

An ideal is nil when each of its elements has some power equal to zero.
The upper nilradical gathers all such ideals into one.
In a commutative ring this is just the set of nilpotent elements.
In a noncommutative ring a nilpotent element need not lie in any nil ideal: in $M_2(\Q)$ the matrix unit $e_{12}$ is nilpotent, but the only nil ideal is $0$.