Timeline for Maximal ideal and Zorn's lemma
Current License: CC BY-SA 2.5
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 28, 2011 at 22:22 | comment | added | Joel David Hamkins | See related question mathoverflow.net/questions/39872/… | |
| Jan 28, 2011 at 19:46 | comment | added | expmat | I understand it might not be a good definition. But still, I am curious about my question. As Dan put it: [is the statement "All rings without infinite, proper, ascending chains have maximal ideals" equivalent to DC?]. | |
| Jan 27, 2011 at 23:57 | comment | added | Joel David Hamkins | I agree, François. | |
| Jan 27, 2011 at 23:41 | comment | added | François G. Dorais | I think the answer to expmat's question is that the ACC is simply not the right definition of Noetherian in plain ZF; the right definition that every nonempty set of ideals has a maximal element. With the right definition, the existence of maximal ideals is trivial. In fact, this is not the only standard theorem about Noetherian rings which is easy to prove in plain ZF with the right definition. | |
| Jan 27, 2011 at 20:40 | comment | added | Qiaochu Yuan | @Willie: agreed. There's nothing there specifically about Noetherian rings, though. | |
| Jan 27, 2011 at 19:50 | comment | added | Willie Wong | These kinds of questions come up so much I am beginning to wonder if consequences.emich.edu/CONSEQ.HTM should not be in the FAQ for MathOverflow. (DC) is form 43, btw, for those interesting in looking. | |
| Jan 27, 2011 at 19:43 | comment | added | Dan Ramras | To rephrase expmat's question, is the statement "All rings without infinite, proper, ascending chains have maximal ideals" equivalent to DC? | |
| Jan 27, 2011 at 19:41 | comment | added | expmat | Thanks, I didn't know about DC but I am still curious as to my final question. Is there a proof of this result that does not use anything beyond ZF axioms? | |
| Jan 27, 2011 at 19:38 | history | answered | Joel David Hamkins | CC BY-SA 2.5 |