---
tag: G00C22AEDH
kind: concept
name: Simple module
msc: [16D60]
sources:
  - url: https://en.wikipedia.org/wiki/Simple_module
same_as:
  wikidata: Q956233
  wikipedia: Simple module
---

## Summary

A simple module is a nonzero [[module|0NZ8JA25WK]] whose only [[submodules|05WDSN597C]] are zero and the module itself.
Simple modules are the building blocks of modules of finite length: the factors of a [[composition series|0XV2A49X6Q]] are simple.
Over a group algebra, simple modules are the same as [[irreducible representations|FX3NEFYFFN]].
Every simple left module over a ring $R$ is isomorphic to $R/\mathfrak{m}$ for a maximal left ideal $\mathfrak{m}$.

## Explanations

### L1 human

A simple module cannot be broken into a smaller nonzero piece inside it.
It plays the role that prime numbers play for integers, or that simple groups play for finite groups.
For the integers, the simple modules are the groups $\Z/p\Z$ with $p$ prime.
Such a group has $p$ elements, and a subgroup has an order dividing $p$, so it is either zero or everything.