# Is every ring the direct limit of Noetherian rings?

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

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## 1 Answer

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.

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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 '11 at 13:21
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 '11 at 14:45
Thanks quid . –  Martin Brandenburg Nov 12 '11 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 '11 at 16:22