---
tag: E2P29GCD9T
kind: concept
name: Indecomposable module
msc: [16D70]
sources:
  - url: https://en.wikipedia.org/wiki/Indecomposable_module
same_as:
  wikidata: Q2499534
  wikipedia: Indecomposable module
---

## Summary

An indecomposable module is a nonzero [[module|0NZ8JA25WK]] that cannot be written as a direct sum of two nonzero submodules.
Every [[simple module|G00C22AEDH]] is indecomposable, but an indecomposable module may have many submodules.
By the Krull–Schmidt theorem, a module of finite length is a direct sum of indecomposable modules in an essentially unique way.
So classifying modules of finite length over a ring comes down to classifying its indecomposable modules.

## Explanations

### L1 human

An indecomposable module cannot be split into two independent nonzero parts.
It may still have smaller pieces inside, but none of them can be split off cleanly.
The group $\Z/4\Z$ is an example: it contains $\set{0, 2}$, but no subgroup is a complement to it.
The group $\Z/6\Z$ is not indecomposable, since it is $\Z/2\Z \oplus \Z/3\Z$.