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 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 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[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