1.6 (Hilbert Basis Theorem). Any ideal $I \subset \mathcal{O}\left(\mathbb{A}^{n}\right)$ is an $\mathcal{O}\left(\mathbb{A}^{n}\right)$-module of finite type, i.e., $I=\left(b_{1}, \ldots, b_{k}\right)$ for some $b_{1}, \ldots, b_{k} \in I$.

Definition 1.5. A module $M$ over a commutative algebra $A$ is said to be of finite type if
$$
\exists b_{1}, \ldots, b_{k} \in M \mid \forall b \in M, \quad b=\sum_{i=1}^{k} a_{i} b_{i} \text { for some } a_{1}, \ldots, a_{k} \in A
$$





Theorem: If $R$ is a polynomial ring in finitely many variables over a field or over the ring of integers, then every ideal in $R$ can be generated by finitely many elements.
Theorem: If a ring $R$ is Noetherian, then the polynomial ring $R[x]$ is Noetherian.

## Proof
![](attachments/Pasted%20image%2020220203135620.png)
![](attachments/Pasted%20image%2020220203135635.png)

For finitely generated algebras:

![](attachments/Pasted%20image%2020220203135730.png)
# Notes

![Pasted image 20220114164712.png](attachments/Pasted%20image%2020220114164712.png)

As a consequence, affine or projective spaces with the Zariski topology are Noetherian topological spaces, which implies that any closed subset of these spaces is compact.