Finitely generated algebra

In mathematics, a finitely generated algebra is an associative algebra A over a field K where there exists a finite set of elements a₁,…,a_n of A such that every element of A can be expressed as a polynomial in a₁,…,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 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[x₁,…,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 then every ideal of A is finitely generated, or equivalently, A is a Noetherian ring.

See also

• Finitely generated module
• Finitely generated field extension