---
tag: WD3G29FZ98
kind: definition
name: Indecomposable module
msc: [16D70]
defines: E2P29GCD9T
uses: [09VDS2W1E8, 0RGZ10DFPG, AWSBZWWMG9, 0X4JKZXJ86]
sources:
  - url: https://en.wikipedia.org/wiki/Indecomposable_module
---

## Definition

Let $R$ be a ring and let $M$ be a left [[$R$-module|09VDS2W1E8]].
The module $M$ is *indecomposable* if:

1. $M \ne 0$;
2. whenever $M = N_1 \oplus N_2$ for [[submodules|0RGZ10DFPG]] $N_1, N_2$ of $M$, one has $N_1 = 0$ or $N_2 = 0$.

## Remarks

Equivalently, $M \ne 0$ and the only [[idempotents|0X4JKZXJ86]] of the [[endomorphism ring|AWSBZWWMG9]] $\End_R(M)$ are $0$ and $1$.
Example: the $\Z$-module $\Z$ is indecomposable.
The check: two nonzero submodules $a\Z$ and $b\Z$ both contain $ab \ne 0$, so their sum is never direct.
Example: $\Z/p^n\Z$ is indecomposable for $p$ prime and $n \ge 1$, since its subgroups form a chain.
Non-example: $\Z/6\Z = \set{0, 3} \oplus \set{0, 2, 4}$ fails clause 2.
Every simple module is indecomposable, but $\Z/4\Z$ is indecomposable and not simple.