Finitely generated algebra

In mathematics, a finitely generated algebra (also called an algebra of finite type) is a commutative 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".
Equivalently, there exist elements formula_1 s.t. the evaluation homomorphism at formula_2
is surjective; thus, by applying the first isomorphism theorem formula_4. 
Conversely, formula_5 for any ideal formula_6 is a formula_7-algebra of finite type, indeed any element of formula_8 is a polynomial in the cosets formula_9 with coefficients in formula_7. Therefore, we obtain the following characterisation of finitely generated formula_7-algebras
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.
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. More precisely, given an affine algebraic set formula_16 we can associate a finitely generated formula_7-algebra
called the affine coordinate ring of formula_19; moreover, if formula_20 is a regular map between the affine algebraic sets formula_16 and formula_22, we can define a homomorphism of formula_7-algebras
then, formula_25 is a contravariant functor from the category of affine algebraic sets with regular maps to the category of reduced finitely generated formula_7-algebras: this functor turns outto be an equivalence of categories
and, restricting to affine varieties (i.e. irreducible affine algebraic sets),
Finite algebras vs algebras of finite type.
We recall that a commutative formula_29-algebra formula_8 is a ring homomorphism formula_31; the formula_29-module structure of formula_8 is defined by
An formula_29-algebra formula_8 is finite if it is finitely generated as an formula_29-module, i.e. there is a surjective homomorphism of formula_29-modules
Again, there is a characterisation of finite algebras in terms of quotients
By definition, a finite formula_29-algebra is of finite type, but the converse is false: the polynomial ring formula_46 is of finite type but not finite.
Finite algebras and algebras of finite type are related to the notions of finite morphisms and morphisms of finite type.