---
tag: N3MRHDKDGC
kind: concept
name: Semisimple module
msc: [16D60]
sources:
  - url: https://en.wikipedia.org/wiki/Semisimple_module
same_as:
  wikidata: Q1485102
  wikipedia: Semisimple module
---

## Summary

A semisimple module is a [[module|0NZ8JA25WK]] that is a direct sum of [[simple modules|G00C22AEDH]].
Equivalently, every submodule of it has a complement, so the module splits as cleanly as a vector space.
Semisimple modules are the modules for which linear algebra works best.
The rings all of whose modules are semisimple are described completely by the Artin–Wedderburn theorem.
By Maschke's theorem, every representation of a finite group over a field of characteristic zero is semisimple.

## Explanations

### L1 human

A semisimple module is built from simple pieces placed side by side, with no gluing between them.
Every vector space is semisimple: it is a direct sum of lines, and each line is simple.
The group $\Z/6\Z$ is semisimple as a $\Z$-module, since it is $\Z/2\Z \oplus \Z/3\Z$.
The group $\Z/4\Z$ is not: its subgroup $\set{0, 2}$ has no complement.