Are there any examples of commutative rings that do not occur as direct limits of Noetherian rings?
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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. 

