Is every ring the direct limit of Noetherian rings? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T01:38:26Z http://mathoverflow.net/feeds/question/48035 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/48035/is-every-ring-the-direct-limit-of-noetherian-rings Is every ring the direct limit of Noetherian rings? Achilleas K 2010-12-02T11:29:42Z 2011-11-12T14:48:02Z <p>Are there any examples of commutative rings that do not occur as direct limits of Noetherian rings?</p> http://mathoverflow.net/questions/48035/is-every-ring-the-direct-limit-of-noetherian-rings/48037#48037 Answer by Martin Brandenburg for Is every ring the direct limit of Noetherian rings? Martin Brandenburg 2010-12-02T11:36:08Z 2011-11-12T14:48:02Z <p>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.</p>