TITLE: Is every ring the direct limit of Noetherian rings?
QUESTION [12 upvotes]: Are there any examples of commutative rings that do not occur as direct limits of Noetherian rings?

REPLY [28 votes]: Every commutative ring is the directed colimit of its subrings that are finitely generated as $\mathbb{Z}$-algebras. The Hilbert Basis Theorem implies that these subrings are noetherian. Actually this method is used in EGA IV, §8.9 to generalize some theorems from noetherian schemes to more general schemes.