---
tag: A72J7ZVNR5
kind: concept
name: Kurosh–Amitsur radical theory
msc: [16N80]
sources:
  - url: https://en.wikipedia.org/wiki/Radical_of_a_ring
---

## Summary

Kurosh–Amitsur radical theory is the general theory of radicals of rings.
A radical in this sense is a class of rings closed under homomorphic images.
Every ring must have a largest ideal in the class, and the quotient by that ideal must have no nonzero ideal in the class.
The [[Jacobson radical|XA0DCNH7WY]], the [[prime radical|WF117EJ2HF]], the [[Levitzki radical|9DD8M9R868]], the [[upper nilradical|3G2JQEYBEK]] and the [[Brown–McCoy radical|EWWP84BGSC]] are all examples.
The theory lets one prove facts about all these radicals at once, and it extends to other algebraic structures.

## Explanations

### L1 human

Many constructions in ring theory pick out a "bad part" of a ring, an ideal one would like to remove.
Removing it should leave a ring with no bad part left.
Radical theory turns this pattern into axioms.
For instance, the class of nil rings, the rings in which every element is nilpotent, is a radical class, and its radical is the largest nil ideal.