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 of A such that every element of A can be expressed as a polynomial in, with coefficients in K. Equivalently, there exist elements such that the evaluation homomorphism at is surjective; thus, by applying the first isomorphism theorem,. Conversely, for any ideal is a K-algebra of finite type, indeed any element of A is a polynomial in the cosets with coefficients in K. Therefore, we obtain the following characterisation of finitely generated K-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.

Examples

Properties

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 we can associate a finitely generated K-algebra called the affine coordinate ring of V; moreover, if is a regular map between the affine algebraic sets and, we can define a homomorphism of K-algebras then, \Gamma is a contravariant functor from the category of affine algebraic sets with regular maps to the category of reduced finitely generated K-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

We recall that a commutative R-algebra A is a ring homomorphism ; the R-module structure of A is defined by An R-algebra A is called finite if it is finitely generated as an R-module, i.e. there is a surjective homomorphism of R-modules Again, there is a characterisation of finite algebras in terms of quotients By definition, a finite R-algebra is of finite type, but the converse is false: the polynomial ring R[X] 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.