 For example, if "R" is Noetherian, then so is the polynomial ring (by Hilbert's basis theorem), any localization "S"−1"R", and also any factor ring "R" / "I".Any non-noetherian ring "R" is the union of its Noetherian subrings.This fact, known as Noetherian approximation, allows the extension of certain theorems to non-Noetherian rings.