---
tag: CZSCNZH8VY
kind: concept
name: Length of a module
msc: ["13C05", "16D10", "13E10"]
sources:
  - url: https://en.wikipedia.org/wiki/Length_of_a_module
same_as:
  wikidata: Q1761084
  wikipedia: Length of a module
---

## Summary

The length of a module is the largest number of strict steps in a chain of its submodules.
It measures the size of a module, as dimension measures the size of a vector space.
A module has finite length exactly when it is both a [[Noetherian module|AK1ZRCXR32]] and an [[Artinian module|YQNZTQGP1X]].
Length is additive along short exact sequences, which makes it a basic tool in dimension theory and multiplicity theory.

## Explanations

### L1 human

Climb from the zero submodule to the whole module, one strict step at a time, and count the steps.
The length is the largest number of steps that you can take.
For a vector space this is its dimension.
For $\Z/12\Z$ as a $\Z$-module the length is $3$, as in $0 \subset 6\Z/12\Z \subset 3\Z/12\Z \subset \Z/12\Z$, matching the three prime factors of $12 = 2 \cdot 2 \cdot 3$.