Timeline for A remark on Cohen's theorem
Current License: CC BY-SA 3.0
5 events
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| Jan 22, 2014 at 1:07 | comment | added | Torsten Schoeneberg | Ah, and now that I come to think of it: Osofsky's ring has Krull dimension 1, whereas all the other counterexamples here have dimension $\ge 2$ -- since a domain of dimension 1 (or any ring of dimension 0) of course can never be a counterexample! | |
| Jan 22, 2014 at 0:47 | comment | added | Neil Epstein | Good point; I hadn't noticed his comment. Actually come to think of it, even many non-discrete valuation rings will work. Let $G_1, \dotsc, G_n$ be subgroups of the reals, with $G_n = Z$ (the integers). Then any valuation ring with value group $G_1 \oplus \cdots \oplus G_n$ has principal maximal ideal, but no such ring is Noetherian unless $n=1$. | |
| Jan 21, 2014 at 23:14 | comment | added | Torsten Schoeneberg | @Neil Epstein: Yes, that generalises Pete L. Clark's comment to the original post. Contrary to Osofsky's ring, those valuation rings are even domains which give counterexamples. | |
| Jan 21, 2014 at 16:16 | comment | added | Neil Epstein | Another source of examples: There are lots of valuation rings where the unique maximal ideal is principal. Any valuation ring that is discrete but not of rank one will do. | |
| Jan 21, 2014 at 10:59 | history | answered | Torsten Schoeneberg | CC BY-SA 3.0 |