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Concept
Finitely generated algebra
Summary
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 a1,...,an of A such that every element of A can be expressed as a polynomial in a1,...,an, with coefficients in K.
Equivalently, there exist elements s.t. the evaluation homomorphism at
is surjective; thus, by applying the first isomorphism theorem, .
Conversely, for any ideal is a -algebra of finite type, indeed any element of is a polynomial in the cosets with coefficients in . Therefore, we obtain the following characterisation of finitely generated -algebras
is a finitely generated -algebra if and only if it is isomorphic to a quotient ring of the type by an ideal .
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.
The polynomial algebra K[x1,...,xn ] is finitely generated. The polynomial algebra in countably infinitely many generators is infinitely generated.
The field E = K(t) of rational functions in one variable over an infinite 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. This is called Zariski's lemma. See also integral extension.
If G is a finitely generated group then the group algebra KG is a finitely generated algebra over K.
A homomorphic 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 over a Noetherian ring then every ideal of A is finitely generated, or equivalently, A is a Noetherian ring.
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