Timeline for Why are principal ideal domains and Dedekind domains prominent, but I always seem to see Noetherian rings rather than Noetherian domains?
Current License: CC BY-SA 2.5
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 2, 2011 at 1:37 | vote | accept | teil | ||
| Dec 19, 2010 at 22:20 | answer | added | Hailong Dao | timeline score: 4 | |
| Dec 19, 2010 at 17:31 | answer | added | Sándor Kovács | timeline score: 1 | |
| Dec 19, 2010 at 7:31 | answer | added | Pete L. Clark | timeline score: 5 | |
| Dec 19, 2010 at 6:28 | comment | added | Alex B. | By the way, examples of principal ideal rings with zero-divisors are also not hard to come by, e.g. $\mathbb{Z}/n\mathbb{Z}$ for composite $n$, or étale algebras over fields, both of which are ubiquitous. | |
| Dec 19, 2010 at 6:12 | answer | added | Emerton | timeline score: 8 | |
| Dec 19, 2010 at 6:09 | comment | added | KConrad | Noetherian domains that are not Dedekind are prominent in algebraic number theory too. In addition to the ring of integers in a number fields, subrings of finite index in the ring of integers are important (they are called orders) and they are not Dedekind if the index is greater than 1. (Example: Z[sqrt(5)].) See Neukirch's Algebraic Number Theory. If you want to develop a general theory of orders, often it's convenient to abstract to the setting of one-dimensional Noetherian domains. | |
| Dec 19, 2010 at 5:13 | comment | added | Alex B. | I think that that's a reflection of where these algebraic concepts are most widely used. Dedekind rings most prominently arise in algebraic number theory as rings of integers and those happen to be integral domains. On the other hand, an extremely important use of Noetherian rings is as coordinate rings of algebraic sets, and those very often have zero divisors, so you need to develop the theory in that generality, rather than limit yourself to domains. | |
| Dec 19, 2010 at 4:53 | history | asked | teil | CC BY-SA 2.5 |