---
tag: WR4WF1K2KW
kind: concept
name: Finitely generated algebra
msc: [08A30, 08A05]
sources:
  - url: https://en.wikipedia.org/wiki/Substructure_(mathematics)
---

## Summary

An [[algebra|QJGPT2W0FT]] is finitely generated when some finite subset of it generates the whole algebra.
Every element can then be built from finitely many fixed elements by repeated use of the operations.
Finitely generated groups, rings and lattices are all special cases.
In the [[subalgebra lattice|T0TNG7AGXT]], the finitely generated subalgebras are exactly the compact elements.

## Explanations

### L1 human

Pick a few elements of an algebra and apply the operations to them again and again.
If every element of the algebra shows up sooner or later, the algebra is finitely generated.
For example, the integers with addition, negation and $0$ are generated by the single element $1$.
Every integer is $1 + 1 + \cdots + 1$, its negative, or $0$.
The rational numbers with the same operations are not finitely generated, because finitely many fractions only reach the multiples of one fixed fraction.