In mathematics, a finitely generated algebra (also called an algebra of finite type) is a commutative associative algebra
where there exists a finite set of elements
can be expressed as a polynomial in
Equivalently, there exist elements
such that the evaluation homomorphism at
is surjective; thus, by applying the first isomorphism theorem,
{\displaystyle A\simeq K[X_{1},\dots ,X_{n}]/{\rm {ker}}(\phi _{\bf {a}})}
for any ideal
-algebra of finite type, indeed any element of
is a polynomial in the cosets
Therefore, we obtain the following characterisation of finitely generated
-algebras[1] 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.
More precisely, given an affine algebraic set
we can associate a finitely generated
-algebra called the affine coordinate ring of
is a regular map between the affine algebraic sets
, we can define a homomorphism of
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), We recall that a commutative
is a ring homomorphism
is called 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] By definition, a finite
-algebra is of finite type, but the converse is false: the polynomial ring
is of finite type but not finite.
-algebra is of finite type and integral, then it is finite.
is a finitely generated
-algebra by a finite number of elements integral over
Finite algebras and algebras of finite type are related to the notions of finite morphisms and morphisms of finite type.