---
tag: Z1045A0V29
kind: concept
name: Uniserial module
msc: [16D70, 16P70]
specializes: [0NZ8JA25WK]
sources:
  - url: https://en.wikipedia.org/wiki/Serial_module
---

## Summary

A uniserial module is a [[module|0NZ8JA25WK]] whose [[submodules|05WDSN597C]] form a chain: of any two submodules, one contains the other.
Uniserial modules are the building blocks of serial modules and [[serial rings|TW3ESQB3GQ]].
Over a [[discrete valuation ring|MGM3CNJW9V]] and over many finite-dimensional algebras, the indecomposable modules of finite length are uniserial, which makes these modules easy to classify.

## Explanations

### L1 human

Picture the submodules of a uniserial module as the rungs of a single ladder.
There are no side branches: you can only go up or down the ladder.
The abelian group $\Z/8\Z$ is uniserial, because its only subgroups are $0 \subset 4\Z/8\Z \subset 2\Z/8\Z \subset \Z/8\Z$.
The group $\Z/6\Z$ is not uniserial, because its subgroups of order $2$ and $3$ do not contain each other.