---
tag: WF117EJ2HF
kind: concept
name: Prime radical
msc: [16N40, 16N60]
sources:
  - url: https://en.wikipedia.org/wiki/Nilradical_of_a_ring
---

## Summary

The prime radical of a ring is the intersection of all its [[prime ideals|E8K2MN5V1E]].
It is also called the lower nilradical or the Baer radical.
For a [[commutative ring|0310REJJY8]] it is the [[nilradical|QQDN6EYQNJ]].
It is the smallest of the classical radicals of a noncommutative ring, and it vanishes exactly for [[semiprime rings|5F0YMCV50R]].

## Explanations

### L1 human

In a commutative ring the nilpotent elements form an ideal, but in a noncommutative ring they need not.
The prime radical is a good replacement for this ideal.
It is a two-sided ideal made of nilpotent elements, cut out by the prime ideals.
For example, in the ring of upper triangular $2 \times 2$ matrices over a field, the prime radical is the set of strictly upper triangular matrices.