---
tag: QFZ3WKEFH6
kind: concept
name: Radical of a module
msc: [16N20, 16D10]
sources:
  - url: https://en.wikipedia.org/wiki/Radical_of_a_module
same_as:
  wikidata: Q7280485
  wikipedia: Radical of a module
---

## Summary

The radical of a module is the intersection of all its maximal submodules.
It is the dual notion of the [[socle|S2KXCAH5MM]], which is a sum of minimal submodules.
For the ring viewed as a module over itself, the radical is the [[Jacobson radical|XA0DCNH7WY]].
It is used to pass from a module to the quotient $M/\operatorname{rad}(M)$ in many arguments, such as Nakayama's lemma.

## Explanations

### L1 human

A maximal submodule is as large as possible without being the whole module.
The radical is what all these largest proper pieces have in common.
Elements of the radical are the "small" or "negligible" part of the module.
For the integers as a $\Z$-module, the maximal submodules are $p\Z$ for primes $p$, and no nonzero integer is divisible by every prime, so the radical is $0$.