---
tag: XA0DCNH7WY
kind: concept
name: Jacobson radical
msc: [13A15, 16N20]
sources:
  - url: https://en.wikipedia.org/wiki/Jacobson_radical
same_as:
  wikidata: Q1677769
  wikipedia: Jacobson radical
---

## Summary

The Jacobson radical of a commutative ring is the intersection of all its [[maximal ideals|09J6KGT65G]].
An element lies in it exactly when $1 - xy$ is a [[unit|019NFGFPR5]] for every $y$ in the ring.
It contains the [[nilradical|QQDN6EYQNJ]], and it is the ideal that appears in [[Nakayama's lemma|15JHBYR6KH]].
For a [[local ring|MZ02BKPHJA]] it is the unique maximal ideal.

## Explanations

### L1 human

The Jacobson radical collects the elements that lie in every maximal ideal.
Such an element is "small" in a precise sense: adding a multiple of it to $1$ never destroys invertibility.
In $\Z$ the maximal ideals are $p\Z$ for the primes $p$, and no nonzero integer is divisible by every prime, so the Jacobson radical of $\Z$ is $\set{0}$.
In $\Z/12\Z$ the maximal ideals are generated by $2$ and $3$, and the Jacobson radical is $\set{0, 6}$.
For noncommutative rings the same name is used for the intersection of the maximal left ideals.