---
tag: 9DD8M9R868
kind: concept
name: Levitzki radical
msc: [16N40]
sources:
  - url: https://en.wikipedia.org/wiki/Radical_of_a_ring
---

## Summary

The Levitzki radical of a ring is its largest locally nilpotent ideal.
An ideal is locally nilpotent when every finite subset of it generates a nilpotent subring.
It lies between the [[prime radical|WF117EJ2HF]] and the [[upper nilradical|3G2JQEYBEK]].
It is one of the classical radicals of the [[Kurosh–Amitsur radical theory|A72J7ZVNR5]].

## Explanations

### L1 human

A nilpotent ideal is one where any long enough product of its elements is zero.
A locally nilpotent ideal asks less: only products of elements chosen from a fixed finite set must vanish when they are long enough.
The Levitzki radical collects all such ideals.
In a commutative ring every nilpotent element generates a nilpotent ideal, and the Levitzki radical is the set of nilpotent elements.