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.
answered Dec 2, 2010 at 11:36
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1$\begingroup$ Should "of its subrings" be replaced by something like "of its finitely-generated subrings"? Surely you meant the compact elements of the lattice of subrings, not the entire lattice? $\endgroup$ Nov 12, 2011 at 13:21
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2$\begingroup$ Not that I am well placed to make such remarks, but I believe at least some would say that (then) 'which' should be 'that' (to signal the restrictive nature of the clause). $\endgroup$– user9072Nov 12, 2011 at 14:45
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$\begingroup$ Oh, it was a linguistic issue, not a mathematical one. You meant "directed colimit of those subrings which are finitely generated as $\mathbb{Z}$-algebras", not "directed colimits of all subrings, which by the way are finitely generated as $\mathbb{Z}$-algebras. $\endgroup$ Nov 12, 2011 at 16:22
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$\begingroup$ @MartinBrandenburg where can I find a copy of EGA IV §8.9? $\endgroup$– ArrowNov 22, 2015 at 11:21
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1$\begingroup$ @Arrow: At your library, perhaps? $\endgroup$ Nov 22, 2015 at 12:14