---
tag: C2H26BYWHE
kind: concept
name: Nilpotent ideal
msc: [16N40]
sources:
  - url: https://en.wikipedia.org/wiki/Nilpotent_ideal
same_as:
  wikidata: Q4321370
  wikipedia: Nilpotent ideal
---

## Summary

A nilpotent ideal of a [[ring|0B1261B5ED]] is a one-sided or two-sided ideal $I$ such that some power $I^n$ is zero.
This means that every product of $n$ elements of $I$ vanishes, a uniform version of being a [[nil ideal|NZ1G9WRWFB]].
Every nilpotent ideal is nil, but a nil ideal need not be nilpotent.
A ring is called semiprime when it has no nonzero nilpotent ideals.
The Jacobson radical of a left Artinian ring is a nilpotent ideal.

## Explanations

### L1 human

In a nil ideal each element dies after enough self-multiplications.
In a nilpotent ideal a single number $n$ works for all of them at once: any product of $n$ elements of the ideal, equal or not, is zero.
In the ring of upper triangular $2 \times 2$ matrices, the matrices with zero diagonal form a nilpotent ideal.
The product of any two of them is zero.