In mathematics, a finitely generated algebra (also called an algebra of finite type-algebra for short, is a commutative associative algebra defined by ring homomorphism, such that every element of -algebra homomorphism.

is a field such that every element of can be expressed as a polynomial, with coefficients in .

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

Conversely for any ideal-algebras:

1. -algebra if and only if it is isomorphic-algebra to a quotient ring

Algebras that are not finitely generated are called infinitely generated.

A finitely generated ring-algebra.

An algebra being finitely generated (of finite type) should not be confused with an algebra being finite (see below). A finite algebra that is finitely generated as a module-algebra defined by ring homomorphism can be expressed as a linear combination being expressible as a polynomial in a finite set of generators in the case of the algebra being finitely generated.

Examples

• The polynomial algebra is finitely generated. The polynomial algebra in countably infinitely many generators is infinitely generated.

• .

• of rational functions in one variable over an infinite field is not a finitely generated algebra over by a single element, , as a field.

• is a finite field extension.

• then the field extension is finite. This is called Zariski's lemma. See also integral extension.

• is a finitely generated group then the group algebra.

• "Quadratic and quaternion algebras are essentially all the associative R-algebras which are finitely generated projective R-modules and have a standard involution."

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 is a finitely generated commutative algebra over a Noetherian ring then every ideal of A is finitely generated, or equivalently, 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-algebra

called the affine coordinate ring is a regular map-algebras

is a contravariant functor from the category-algebras: this functor turns out to be an equivalence of categories

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

Finite algebras vs algebras of finite type

-algebra is a ring homomorphism-module is defined by

is called finite if it is finitely generated-modules

Again, there is a characterisation of finite algebras in terms of quotients:

1. -submodule.

-algebra is of finite type and integral.

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

References

See also

• Finitely generated module

• Finitely generated field extension

• Artin–Tate lemma

• Noether normalization lemma

• Finite algebra

• Morphism of finite type