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},...,*a*_{n} of *A* such that every element of *A* can be expressed as a polynomial in *a*_{1},...,*a*_{n}, with coefficients in *K*.

Equivalently, there exist elements s.t. the evaluation homomorphism at

is surjective; thus, by applying the first isomorphism theorem .

Conversely, for any ideal is a -algebra of finite type, indeed any element of is a polynomial in the cosets with coefficients in . Therefore, we obtain the following characterisation of finitely generated -algebras^{[1]}

- is a finitely generated -algebra if and only if it is isomorphic to a quotient ring of the type by an ideal .

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},...,*x*_{n}] is finitely generated. The polynomial algebra in infinitely countably 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 ring*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 we can associate a finitely generated -algebra

called the affine coordinate ring of ; moreover, if is a regular map between the affine algebraic sets and , we can define a homomorphism of -algebras

then, is a contravariant functor from the category of affine algebraic sets with regular maps to the category of reduced finitely generated -algebras: this functor turns out^{[2]}to be an equivalence of categories

and, restricting to affine varieties (i.e. irreducible affine algebraic sets),

## Finite algebras vs algebras of finite type

We recall that a commutative -algebra is a ring homomorphism ; the -module structure of is defined by

An -algebra is * finite* if it is finitely generated as an -module, i.e. there is a surjective homomorphism of -modules

Again, there is a characterisation of finite algebras in terms of quotients^{[3]}

- An -algebra is finite if and only if it is isomorphic to a quotient by an -submodule .

By definition, a finite -algebra is of finite type, but the converse is false: the polynomial ring is of finite type but not finite.

Finite algebras and algebras of finite type are related to the notions of finite morphisms and morphisms of finite type.

## References

**^**Kemper, Gregor (2009).*A Course in Commutative Algebra*. Springer. p. 8. ISBN 978-3-642-03545-6.**^**Görtz, Ulrich; Wedhorn, Torsten (2010).*Algebraic Geometry I. Schemes With Examples and Exercises*. Springer. p. 19. ISBN 978-3-8348-0676-5.**^**Atiyah, Michael Francis; MacDonald, Ian Grant (1994).*Introduction to commutative algebra*. CRC Press. p. 21. ISBN 9780201407518.

## See also

- Finitely generated module
- Finitely generated field extension
- Artin–Tate lemma
- Finite algebra
- Morphisms of finite type