Finitely generated algebra
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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 a1,…,an 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.
[edit] Examples
- The polynomial algebra K[x1,…,xn] 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.
[edit] 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.
