Hilbert Basis Theorem

A theorem stating that if R is a Noetherian ring (commutative with unity, typically), then the polynomial ring R[x] is also Noetherian; by induction, R[x1,...,xn] is Noetherian for every finite n. Equivalently, ideals in such polynomial rings are finitely generated.

A theorem asserting that if R is a Noetherian commutative ring then the polynomial ring R[x1,…,xn] in finitely many indeterminates over R is also Noetherian; equivalently, every ideal of R[x1,…,xn] is finitely generated when R satisfies the ascending chain condition on ideals.