Global InformationFinitely generated algebra information
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 such that the evaluation homomorphism at
![{\displaystyle \phi _{\bf {a}}\colon K[X_{1},\dots ,X_{n}]\twoheadrightarrow A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0ee219e7577c42769cb8ef81acfb6561269bfc19)
is surjective; thus, by applying the first isomorphism theorem, .
![{\displaystyle A\simeq K[X_{1},\dots ,X_{n}]/{\rm {ker}}(\phi _{\bf {a}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/caafa80ab6d61e0759ffb962d05844c7600cbcb8)
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[1]
![{\displaystyle A:=K[X_{1},\dots ,X_{n}]/I}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ac7b561ff0f8006c91e2bfb03f1d4bbcb4928acf)
![{\displaystyle I\subseteq K[X_{1},\dots ,X_{n}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5faa838f465c222f486b0bcc7944d2e84cd3f3c7)
- is a finitely generated -algebra if and only if it is isomorphic to a quotient ring of the type by an ideal .
![{\displaystyle K[X_{1},\dots ,X_{n}]/I}](https://wikimedia.org/api/rest_v1/media/math/render/svg/304a803b86c140d120956a3aae51d7000480c89b)
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.
- ^ Kemper, Gregor (2009). A Course in Commutative Algebra. Springer. p. 8. ISBN 978-3-642-03545-6.