---
tag: EWWP84BGSC
kind: concept
name: Brown–McCoy radical
msc: [16N80]
sources:
  - url: https://en.wikipedia.org/wiki/Radical_of_a_ring
---

## Summary

The Brown–McCoy radical of a ring with identity is the intersection of its maximal two-sided ideals.
It is the two-sided analogue of the [[Jacobson radical|XA0DCNH7WY]], which uses maximal left ideals instead.
It contains the Jacobson radical, and for a [[commutative ring|0310REJJY8]] the two radicals coincide.
It is a radical in the sense of the [[Kurosh–Amitsur radical theory|A72J7ZVNR5]].

## Explanations

### L1 human

The quotients of a ring by its maximal two-sided ideals are the simple rings it maps onto.
The Brown–McCoy radical is what all these maps kill.
For the integers, the maximal ideals are $p\Z$ for primes $p$, and their intersection is $0$.
Consider the ring of all linear maps of a countably infinite-dimensional vector space.
Its Brown–McCoy radical is the ideal of maps of finite rank, although its Jacobson radical is $0$.