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
For finitely generated algebras:
# Notes
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.