# Finitely generated algebra

In mathematics, a finitely generated algebra (also called an algebra of finite type) is a commutative associative algebra $A$ over a field $K$ where there exists a finite set of elements $a_{1},\dots ,a_{n}$ of $A$ such that every element of $A$ can be expressed as a polynomial in $a_{1},\dots ,a_{n}$, with coefficients in $K$.
Equivalently, there exist elements $a_{1},\dots ,a_{n}\in A$ such that the evaluation homomorphism at ${\bf {a}}=(a_{1},\dots ,a_{n})$
    $\phi _{\bf {a}}\colon K[X_{1},\dots ,X_{n}]\twoheadrightarrow A$
is surjective; thus, by applying the first isomorphism theorem, $A\simeq K[X_{1},\dots ,X_{n}]/{\rm {ker}}(\phi _{\bf {a}})$.
Conversely, $A:=K[X_{1},\dots ,X_{n}]/I$ for any ideal $I\subseteq K[X_{1},\dots ,X_{n}]$ is a $K$-algebra of finite type, indeed any element of $A$ is a polynomial in the cosets $a_{i}:=X_{i}+I,i=1,\dots ,n$ with coefficients in $K$. Therefore, we obtain the following characterisation of finitely generated $K$-algebras
    $A$ is a finitely generated $K$-algebra if and only if it is isomorphic as a $K$-algebra to a quotient ring of the type $K[X_{1},\dots ,X_{n}]/I$ by an ideal $I\subseteq K[X_{1},\dots ,X_{n}]$.
If it is necessary to emphasize the field K then the algebra is said to be finitely generated over K. Algebras that are not finitely generated are called infinitely generated.

## Examples

* The polynomial algebra $K[x_{1},\dots ,x_{n}]$ is finitely generated. The polynomial algebra in countably infinitely many generators is infinitely generated.
* The field $E=K(t)$ of rational functions in one variable over an infinite field $K$ is not a finitely generated algebra over $K$. On the other hand, $E$ is generated over $K$ by a single element, $t$, as a field.
* If $E/F$ is a finite field extension then it follows from the definitions that $E$ is a finitely generated algebra over $F$.
* Conversely, if $E/F$ is a field extension and $E$ is a finitely generated algebra over $F$ then the field extension is finite. This is called Zariski's lemma. See also integral extension.
* If $G$ is a finitely generated group then the group algebra $KG$ is a finitely generated algebra over $K$.

## Properties

* A homomorphic image of a finitely generated algebra is itself finitely generated. However, a similar property for subalgebras does not hold in general.
* Hilbert's basis theorem: if A is a finitely generated commutative algebra over a Noetherian ring then every ideal of A is finitely generated, or equivalently, A is a Noetherian ring.

## Relation with affine varieties

Finitely generated reduced commutative algebras are basic objects of consideration in modern algebraic geometry, where they correspond to affine algebraic varieties; for this reason, these algebras are also referred to as (commutative) affine algebras. More precisely, given an affine algebraic set $V\subseteq \mathbb {A} ^{n}$ we can associate a finitely generated $K$-algebra
    $\Gamma (V):=K[X_{1},\dots ,X_{n}]/I(V)$
called the affine coordinate ring of $V$; moreover, if $\phi \colon V\to W$ is a regular map between the affine algebraic sets $V\subseteq \mathbb {A} ^{n}$ and $W\subseteq \mathbb {A} ^{m}$, we can define a homomorphism of $K$-algebras
    $\Gamma (\phi )\equiv \phi ^{*}\colon \Gamma (W)\to \Gamma (V),\,\phi ^{*}(f)=f\circ \phi ,$
then, $\Gamma$ is a contravariant functor from the category of affine algebraic sets with regular maps to the category of reduced finitely generated $K$-algebras: this functor turns out to be an equivalence of categories
    $\Gamma \colon ({\text{affine algebraic sets}})^{\rm {opp}}\to ({\text{reduced finitely generated }}K{\text{-algebras}}),$
and, restricting to affine varieties (i.e. irreducible affine algebraic sets),
    $\Gamma \colon ({\text{affine algebraic varieties}})^{\rm {opp}}\to ({\text{integral finitely generated }}K{\text{-algebras}}).$

## Finite algebras vs algebras of finite type

We recall that a commutative $R$-algebra $A$ is a ring homomorphism $\phi \colon R\to A$; the $R$-module structure of $A$ is defined by
    $\lambda \cdot a:=\phi (\lambda )a,\quad \lambda \in R,a\in A.$
An $R$-algebra $A$ is called finite if it is finitely generated as an $R$-module, i.e. there is a surjective homomorphism of $R$-modules
    $R^{\oplus _{n}}\twoheadrightarrow A.$
Again, there is a characterisation of finite algebras in terms of quotients
    An $R$-algebra $A$ is finite if and only if it is isomorphic to a quotient $R^{\oplus _{n}}/M$ by an $R$-submodule $M\subseteq R$.
By definition, a finite $R$-algebra is of finite type, but the converse is false: the polynomial ring $R[X]$ is of finite type but not finite. However, if an $R$-algebra is of finite type and integral, then it is finite. More precisely, $A$ is a finitely generated $R$-module if and only if $A$ is generated as an $R$-algebra by a finite number of elements integral over $R$.
Finite algebras and algebras of finite type are related to the notions of finite morphisms and morphisms of finite type.