# Finitely generated algebra

In mathematics, a **finitely generated algebra** (also called an **algebra of finite type**) is an 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*. 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**. 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**.

## 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.

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