Hilbert basis theorem

If a commutative ring is Noetherian, then its polynomial ring in finitely many variables is Noetherian.

Hilbert basis theorem

Hilbert basis theorem: Let $R$ be a commutative ring . If $R$ is Noetherian (i.e. every ascending chain of ideals stabilizes), then the polynomial ring $R[x]$ is Noetherian. More generally, $R[x_1,\dots,x_n]$ is Noetherian for every $n\ge 1$ .