In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a finitely generated algebra is an associative algebra

Associative algebra

In mathematics, an associative algebra A is an associative ring that has a compatible structure of a vector space over a certain field K or, more generally, of a module over a commutative ring R...

 A over a field

Field (mathematics)

In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...

 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

Polynomial

In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...

 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 commutative algebras

Commutative ring

In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....

 are basic objects of consideration in modern algebraic geometry

Algebraic geometry

Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...

, 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 countably many generators is infinitely generated.
• The field E = K(t) of rational function
  Rational function
  In mathematics, a rational function is any function which can be written as the ratio of two polynomial functions. Neither the coefficients of the polynomials nor the values taken by the function are necessarily rational.-Definitions:...
  
  s in one variable over a given 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, see integral extension.
• If G is a finitely generated group then the group ring
  Group ring
  In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring and its basis is one-to-one with the given group. As a ring, its addition law is that of the free...
  
   KG is a finitely generated algebra over K.

Properties

• A homomorphic image
  Homomorphism
  In abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures . The word homomorphism comes from the Greek language: ὁμός meaning "same" and μορφή meaning "shape".- Definition :The definition of homomorphism depends on the type of algebraic structure under...
  
   of a finitely generated algebra is itself finitely generated. However, a similar property for subalgebra
  Subalgebra
  In mathematics, the word "algebra", when referring to a structure, often means a vector space or module equipped with an additional bilinear operation. Algebras in universal algebra are far more general: they are a common generalisation of all algebraic structures...
  
  s does not hold in general.
• Hilbert's basis theorem
  Hilbert's basis theorem
  In mathematics, specifically commutative algebra, Hilbert's basis theorem states that every ideal in the ring of multivariate polynomials over a Noetherian ring is finitely generated. This can be translated into algebraic geometry as follows: every algebraic set over a field can be described as the...
  
  : if A is a finitely generated commutative algebra then every ideal of A is finitely generated, or equivalently, A is a Noetherian ring
  Noetherian ring
  In mathematics, more specifically in the area of modern algebra known as ring theory, a Noetherian ring, named after Emmy Noether, is a ring in which every non-empty set of ideals has a maximal element...
  
  .

See also

• Finitely generated module
• Finitely generated field extension