Timeline for Exotic principal ideal domains
Current License: CC BY-SA 2.5
6 events
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| Feb 24, 2011 at 14:57 | comment | added | Emil Jeřábek | @Pete: No. PIDs have additional first-order properties that do not hold for every Bézout domain, for example: every non-unit is divisible by a prime. BTW, being a PID is not a first-order property, but it's not that bad: it's expressible in $L_{\omega_1\omega}$. | |
| Feb 24, 2011 at 14:50 | comment | added | Pete L. Clark | @Qiaochu: right. The corresponding first order property is that every finitely generated ideal is principal -- i.e., Bezout domains. In other words, every ultraproduct of Bezout domains is Bezout. Off the top of my head, I wonder whether every Bezout domain is an ultraproduct of PIDs? | |
| Feb 24, 2011 at 14:17 | comment | added | Qiaochu Yuan | Oh, I see. Being a PID is not a first-order property. My mistake. | |
| Feb 24, 2011 at 13:45 | comment | added | Emil Jeřábek | Not really. In almost any nontrivial case, an ultraproduct will contain an infinite chain of divisors, and thus fail to be a UFD. | |
| Feb 24, 2011 at 13:38 | comment | added | Qiaochu Yuan | Nice. I suppose I can also take ultraproducts of PIDs to get more exotic ones... | |
| Feb 24, 2011 at 12:38 | history | answered | Emil Jeřábek | CC BY-SA 2.5 |