---
tag: AC5TFFQT9M
kind: concept
name: Finite type algebra
msc: ["13E15", 13B02]
specializes: [DPBJ04JMH8]
sources:
  - url: https://en.wikipedia.org/wiki/Finitely_generated_algebra
same_as:
  wikidata: Q2835968
  wikipedia: Finitely generated algebra
---

## Summary

An [[algebra|DPBJ04JMH8]] $B$ over a commutative ring $A$ is of finite type if finitely many elements of $B$ generate it as an $A$-algebra.
Equivalently, $B$ is a quotient of a [[polynomial ring|0SDC934R52]] $A[x_1, \ldots, x_n]$ in finitely many variables.
Finitely generated algebras over a field are the coordinate rings of affine varieties, the main objects of classical algebraic geometry.
Over a Noetherian base they are Noetherian, by [[Hilbert's basis theorem|0YKA9J3XEE]].

## Explanations

### L1 human

An algebra is of finite type if every element can be written as a polynomial, with coefficients in $A$, in a fixed finite list of elements.
The ring $\C[x, y]/(x^2 + y^2 - 1)$ of polynomial functions on a circle is generated over $\C$ by the two coordinate functions.
Finite type is weaker than finite: $k[x]$ is of finite type over $k$, generated by $x$, but it is not a finitely generated $k$-module.
The ring $\Q$ is not of finite type over $\Z$: finitely many fractions only produce finitely many prime denominators.