Finitely generated algebra
- Finitely generated algebra
In mathematics, a finitely generated algebra is an associative algebra "A" over a field "K" such that every element of "A" can be expressed as a polynomial in a finite set of elements "a"1,…,"a"n of "A", 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 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.
Examples
* The polynomial algebra "K" ["x"1,…,"x"n] is finitely generated. The polynomial algebra in countably many generators is infinitely generated.
* The field "E" = "K"("t") of rational functions 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 "KG" is a finitely generated algebra over "K".
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: if "A" is a finitely generated commutative algebra then every ideal of "A" is finitely generated, or equivalently, "A" is a Noetherian ring.
See also
* Finitely generated module
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