Timeline for When can we prove constructively that a ring with unity has a maximal ideal?
Current License: CC BY-SA 2.5
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 27, 2011 at 23:04 | comment | added | Joel David Hamkins | More generally, in any atomless Boolean algebra, there are no principal ultrafilters. | |
| Nov 28, 2009 at 21:34 | comment | added | Guillermo Mantilla | @ Buzzard: you are 100% right, I should have said non-principal ones or as Greg mentioned ultrafilters extending the Fr\'echet filter. | |
| Nov 28, 2009 at 17:52 | comment | added | Greg Kuperberg | According to this authority, that's correct. at.yorku.ca/cgi-bin/… The existence of ultrafilters is also supposedly weaker than the axiom of choice. | |
| Nov 28, 2009 at 17:26 | comment | added | Kevin Buzzard | Yes good point Greg. So there's a very natural candidate for a ring that might well have no maximal ideals if AC fails sufficiently badly. My memory is that it's standard that there are models of ZF where the only ultrafilters on Z are the principal ones. | |
| Nov 28, 2009 at 17:11 | comment | added | Greg Kuperberg | Of these Boolean algebras. Consider instead the quotient of $\Omega(\mathbb{N})$ by the ideal of finitely supported Boolean functions. Any maximal ideal of that is a non-principal ultrafilter. | |
| Nov 28, 2009 at 14:30 | comment | added | Kevin Buzzard | The existence of ultrafilters is easy: take a principal one. It's the non-principal ones that need AC. So one can find maximal ideals of Boolean algebras constructively, just not interesting ones ;-) | |
| Nov 28, 2009 at 10:38 | history | answered | Guillermo Mantilla | CC BY-SA 2.5 |