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

flag	asked Dec 2 2010 at 11:29 Achilleas K 248●2●7

Martin Brandenburg · Accepted Answer · 2011-11-12 14:48:02Z UTC

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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.

edited Nov 12 2011 at 14:48

answered Dec 2 2010 at 11:36

Martin Brandenburg
29k●2●45●129

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? – Andrej Bauer Nov 12 2011 at 13:21

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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). – quid Nov 12 2011 at 14:45

Thanks quid . – Martin Brandenburg Nov 12 2011 at 14:48

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. – Andrej Bauer Nov 12 2011 at 16:22

Is every ring the direct limit of Noetherian rings?

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