Timeline for Can we prove that the ring of formal power series over a noetherian ring is noetherian without axiom of choice?
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| Apr 13, 2017 at 12:19 | history | edited | CommunityBot |
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| Jan 21, 2015 at 5:09 | answer | added | Pace Nielsen | timeline score: 5 | |
| Jan 14, 2014 at 3:00 | comment | added | François G. Dorais | In my comment David links to, I say that the maximal element definition is "the right definition" in the absence of choice. There are nuances here and context matters. The maximal element definition is arguably "the right definition" for ZF. Emil's proposal is also "the right definition" but in different settings, such as constructive mathematics. I'm not sure the ACC definition is "the right definition" in any context but it is a very useful one in ZFC. | |
| Jan 14, 2014 at 2:52 | comment | added | François G. Dorais | Side remark: all definitions proposed here are distinct as shown by Hodges, Six impossible rings, J. Algebra 31 (1974), 218–244; MR0347814. | |
| Jan 14, 2014 at 1:34 | comment | added | Joel David Hamkins | The dispute is similar to the difference between well-foundedness as every-nonempty-set-has-minimal-element vs. no-descending-sequences, but in the absence of AC, it is usually the former that is considered the more robust concept, and it is this that corresponds to the OP's definition here. | |
| Jan 14, 2014 at 1:26 | comment | added | David E Speyer | @EmilJeřábek For example, this definition implies that every ideal is contained in a maximal ideal. We've had a bunch of questions about this over the years, e.g. mathoverflow.net/questions/7025 and mathoverflow.net/questions/53523/maximal-ideal-and-zorns-lemma and most of the logicians seem to agree that the OP's definition is better in the absence of choice. For example, see Francois Dorais' comment on the second question I linked. | |
| Jan 14, 2014 at 0:38 | review | Close votes | |||
| Jan 14, 2014 at 3:06 | |||||
| Jan 14, 2014 at 0:25 | comment | added | Emil Jeřábek | Is there a motivation for using this particular definition of a noetherian ring? I would have thought that in absence of choice, the more natural definition is that every ideal is finitely generated (or equivalently, every directed set of ideals has a largest element). I believe that the standard proof that $A$ noetherian implies $A[[x]]$ noetherian goes through without choice under the “finitely generated” definition. | |
| Jan 13, 2014 at 23:56 | history | edited | Makoto Kato | CC BY-SA 3.0 |
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| Jan 13, 2014 at 23:47 | history | asked | Makoto Kato | CC BY-SA 3.0 |