Talk:Dedekind domain

Clearer descriptions
Could we have a clearer description of the algebraic integer examples? They seem to be the most important examples, and historically the motivating ones. We start with a finite extension of Q and then consider the ring of integral elements over Q, is that always Dedekind? 156.99.90.179 21:14, 14 November 2002 (UTC)


 * you mast consider the ring of integral elements over Z and then it always Dedekind and yas it is the most important example(28.3.2006) — Preceding unsigned comment added by 212.25.124.149 (talk) 15:09, 28 March 2006 (UTC)




 * It's never Dedekind, regardless of whether it's always or sometimes a Dedekind domain.


 * Simimarly, no sequence is ever Cauchy, regardless of whether it's a Cauchy sequence.


 * The phrase "Cauchy sequence" is a compound; "Cauchy" is not an adjective.


 * — Preceding unsigned comment added by 131.183.81.100 (talk) 23:13, 14 November 2002 (UTC)


 * I've commonly heard expressions such as "if the sequence is Cauchy...". It may have been initially a compound, but it has evolved, clearly. Since the term is exclusive to the domain of mathematics, I suppose it is up to those that use it to define its correct grammar. -- Tarquin 00:28, 15 November 2002 (UTC)


 * I hear that expression all the time too, but doesn't it sound rather uncouth once you've thought of that issue? -- Mike Hardy — Preceding unsigned comment added by 131.183.81.100 (talk) 21:55, 16 November 2002 (UTC)


 * 
 * Grammar is defined by usage. Read a paper: "The metric is Hodge." "The sequence is Cauchy." "The ring is Dedekind." It's either that or we're doomed to horrible adjectives such as "Cauchian," "Hodgian," and "Dedekindian", God forbid. It doesn't sound uncouth at all. It sounds far better than barbarizing the names of the brilliant.


 * alodyne — Preceding unsigned comment added by Alodyne (talk • contribs) 06:23, 26 November 2002 (UTC)


 * "A concrete example is the set {a&#8730;2 + bi + c : a, b, c in Z }, considered as a subring of C."


 * This surely is not a Dedekind domain, since it is not a ring. Should we include "di&#8730;2" or remove one of "a&#8730;2" or "bi"? --217.228.232.254 23:50, 4 March 2005 (UTC)


 * How can I get a sketch of the proof of the following theorem: every ideal can be uniquely factored as a product of prime ideals? — Preceding unsigned comment added by 212.25.124.149 (talk) 14:50, 28 March 2006 (UTC)


 * Here's a list of lemmas and theorems, unfortunately without proofs: http://www.math.niu.edu/~beachy/aaol/commutative.html#12101 AxelBoldt 22:45, 28 March 2006 (UTC)

This page defines a Dedekind domain as having Krull dimension 1. However, that would exclude fields (as you proceed to state), but then you also state as an example that all PID (principle ideal domains) are Dedekind domains. This is wrong under your definition, since e.g. every field is a PID but has Krull dimension 0.

The commonly found definition (see e.g. http://planetmath.org/encyclopedia/DedekindDomain.html or http://www.mathreference.com/id-dd,intro.html, but also "Commutative Algerbra" by Zariski&Samuels.) permits fields, and basically requires the dimension to be *at most* 1. Oh and of course the original definition is altogether different, but I digress :). Hence either the example or the definition given on this page should be changed. I would prefer adjusting the definition, as that is what I find virtually everywhere in the literature, but I am wary of making such a big change myself, since I am relatively new to editing Wikipedia :). BlackFingolfin 11:26, 18 April 2006 (UTC)


 * Thanks a lot for finding this mistake. It's funny that even the venerable Encyclopaedia of Mathematics makes the same mistake (http://eom.springer.de/d/d030550.htm).
 * You could have just jumped in and made the change: Be bold in updating pages. If somebody doesn't like your change, they'll let you know soon enough :-) Cheers and have fun around here, because that's what it's all about. AxelBoldt 21:23, 22 April 2006 (UTC)

Major revision begun
Speaking of bold, I began a major revision of this entry. It seemed to me that the previous version was a bit anemic all around, and in particular did not really emphasize the difference between PIDs and Dedekind domains.

What I have just done is very incomplete. In particular I would like a discussion of the following topics: (i) fractional ideals; (ii) the ideal class group of a Dedekind domain, including viewing it as a special case of the Picard group construction; (iii) some results on module theory over a Dedekind domain, in particular the isomorphism from the class group to reduced K_0. (iv) some other miscellaneous results: e.g. CRT, 1+epsilon generation of ideals, and so forth. Also references will be included!

If you have ideas for what might be included, please do post something here. I don't know everything about Dedekind domains by any means (whatever that might mean!), but I am teaching a graduate course in algebraic number theory so have already looked up lots of results and have things pretty firmly in mind. Plclark (talk) 02:54, 27 February 2008 (UTC)


 * I have made a second round of additions, enough so that my third wave should probably be to
 * go back and insert proper references and check formatting rather than introduce more material.
 * Still, I welcome suggestions about what other aspects might be included. Plclark (talk) 06:44, 24 April 2008 (UTC)

"unexplained link removal"
I thought it was self-evident, but apparently not. The Manual of Style frowns on wikilinks to an anchor. If you want to refer to that list there, then you should move it up in the text.

It's a small style point, but it contributes to the article's overall illegibility. Elinruby (talk) 16:25, 12 May 2021 (UTC)
 * It is evident that most readers prefer a direct link instead searching in the whole article the part to which "below" refers. So, there is nothing "self-evident" in the reason for removing this link. If the Manual of Style discourages such a link, please, provide a link to the relevant section. D.Lazard (talk) 19:54, 12 May 2021 (UTC)


 * Also, when you refer to the Manual of Style, please refer to the specific section that applies, preferably with a link to it. —Anita5192 (talk) 19:56, 12 May 2021 (UTC)