---
tag: 0XHM30WBVB
kind: definition
name: Finitely generated module
msc: ["13E15", 16D10]
defines: 05QY1VEBYD
uses: [09VDS2W1E8]
sources:
  - url: https://en.wikipedia.org/wiki/Finitely_generated_module
    locator: Definition
---

## Definition

Let $R$ be a ring with identity element and let $M$ be a left [[module|0NZ8JA25WK]] over $R$.
The module $M$ is *finitely generated* when there are $x_1, \dots, x_n \in M$ such that each $x \in M$ is
$$x = r_1 x_1 + \cdots + r_n x_n \qquad \text{for some } r_1, \dots, r_n \in R.$$

## Remarks

Over a field, the finitely generated modules are the finite-dimensional vector spaces.
$\Z^n$ and $\Z / n\Z$ are finitely generated over $\Z$; the group $\Q$ is not.
Over a Noetherian ring, each submodule of a finitely generated module is finitely generated.
The finitely generated modules over a principal ideal domain are classified by the structure theorem.

## Explanations

### L1 human

A module is finitely generated when finitely many elements produce all others by linear combinations.
This is the module analogue of a finite-dimensional vector space.
Unlike in vector spaces, the generators need not be independent.