---
tag: V1H5X4H8Q7
kind: definition
name: Radical of a module
msc: [16N20, 16D10]
defines: QFZ3WKEFH6
uses: [0NZ8JA25WK, 05WDSN597C, XA0DCNH7WY]
sources:
  - url: https://en.wikipedia.org/wiki/Radical_of_a_module
---

## Definition

Let $R$ be a ring and let $M$ be a left [[module|0NZ8JA25WK]] over $R$.
A *maximal submodule* of $M$ is a [[submodule|05WDSN597C]] $N \ne M$ such that no submodule lies strictly between $N$ and $M$.
The *radical* of $M$, written $\operatorname{rad}(M)$, is the intersection of all maximal submodules of $M$.
If $M$ has no maximal submodule, then $\operatorname{rad}(M) = M$.

## Remarks

Example: let $M = \Z/12\Z$ as a $\Z$-module.
Its maximal submodules are $2\Z/12\Z$ and $3\Z/12\Z$, so $\operatorname{rad}(M) = 6\Z/12\Z$, which has $2$ elements.

Non-example of a proper radical: the $\Z$-module $\Q$ has no maximal submodule, because every quotient $\Q/N$ with $N \ne \Q$ is divisible and so is not simple.
So $\operatorname{rad}(\Q) = \Q$.

For $M = R$ as a left module over itself, $\operatorname{rad}(R)$ is the [[Jacobson radical|XA0DCNH7WY]] $J(R)$.
Equivalently, $\operatorname{rad}(M)$ is the intersection of the kernels of all module homomorphisms from $M$ to simple modules.
For every $M$ one has $J(R) M \subseteq \operatorname{rad}(M)$, with equality when $R$ is left Artinian.