Talk:Bézout domain

Additions

I made some changes to this page -- mostly additions -- and mostly taken out of Kaplansky's book on Commutative rings. I removed the stub description, because I do not know of anything specific that the article is lacking.

Probably others will want to do some minor reformatting and other cosmetic changes. Plclark (talk) 10:05, 6 April 2008 (UTC)

Examples

I just noticed that the claim that the two examples cited in the lead are the "best known" examples comes from P.M. Cohn's article, which is referenced. I now understand better why some other editors put this back after I deleted it the first time, and I apologize for not following up on the link before.

Nevertheless, I still feel that this is unnecessary and poorly justified POV language. If this sentiment must be expressed, a more neutral and factual way would be "In 1967, the distinguished algebraist Paul Moritz Cohn wrote that the best known...." But I don't think it's necessary. Why argue about whether some examples are better known than others?

But, if we are, I would say that nondiscrete valuation rings are significantly more ubiquitous examples of non-Noetherian Bezout domains. Certainly the example of valuation rings should be added to the article in some form. (As a general rule, I don't necessarily think that the lead needs to contain examples, but maybe that is my inner Bourbakiste speaking.) Plclark (talk) 12:58, 17 June 2009 (UTC)

Valuation rings are mentioned, but much lower in the article. I can dig up a few other sources that also say that holomorphic functions (on riemann surfaces) and the ring of algebraic integers are the "best known" examples, but I don't personally have any strong preference between "best known" or "well known" or just "some". In my opinion the examples would not be in the lead unless they were the "best" anyways, so stating that they are the best known might be redundant. JackSchmidt (talk) 13:12, 17 June 2009 (UTC)

I found the material on valuation rings and then remembered that it was I who added it. Maybe a bit of reorganization would be useful. Anyway, I think the two examples above are "good enough" to be in the lead.

But as an aside, let me say that I am an arithmetic geometer / number theorist with interests in commutative algebra, and the fact that the ring of all algebraic integers is Bezout, although known to me via Kaplansky's book, is not something that I have ever seen used in my field. I suspect complex geometers would say something similar about rings of holomorphic functions. In contrast, valuation rings and their structural properties are ubiquitous in algebra and geometry. Plclark (talk) 14:55, 17 June 2009 (UTC)

I too disagree with the usage of "best known" and hence I replaced it with "well known" (although I do not think either should really be emphasized). For instance, one might say that the "best known" examples of topological spaces are manifolds; this is a correct usage. On the other hand, one should not interpret this as "the most interesting examples of topological spaces are manifolds," which is false. An analogue of this applies here. When using "best known" one must be careful not to suggest "most interesting" and I think therefore that such language should be avoided in most situations. --PST 02:44, 18 June 2009 (UTC)

I like JackSchmidt's argument a lot: if examples are obscure or of limited interest, why do we mention them at all in the lede, the most prominent part of the article? This makes the issue moot, but I would like to make some of my own arguments too. First, Plclark's argument (and PST's similar one) makes sense only if we're writing a textbook or lecture. The purpose of the lede, well at least the primary purpose, is not to motivate the subject so to entice the readers to the remainder of the article, but to establish the context (in addition to being a summary of the article). Examples that get mentioned in the lede don't have to be interesting, important nor widely studied. In other words, the purpose of the lede is not equivalent to, say, that of the introductory part of chapters of textbooks. That is, if we were, say, introducing Bezout domain in an algebra text, maybe emphasis should be on valuation rings, where applications are. (Disclosure: as it might be clear, I don't know much about Bezout domains. I do remember Kaplansky mentioned the ring of algebraic integers as an example of a Bezout domain in his commutative ring text, though.). But that is not the case here. Second, PST's argument. Yes, "best known" doesn't imply "most interesting", and I think we can count that the readers know this. In fact, we use the language "best known" all over Wikipedia: most articles on, for instance, actors, mention which roles they are best known, no matter their acting are dull or uninteresting. -- Taku (talk) 21:19, 18 June 2009 (UTC)

My primary view of Wikipedia is that it is not "formulaic"; that is, the lead does not need to be written for the purpose of summarizing the article. On the other hand, as Taku points out, the lead does establish context.

With this in mind, mathematics is a different subject to all others. Often, writing a good mathematics article can be tricky because one must mention several facts about the concept in question. Unlike some other subjects, a list of "facts" about Bézout domains does not constitute an article. Furthermore (perhaps somewhat blunt), I do not think I care much for that which P.M. Cohn comments on based on his opinion (although I do not question is reputability). Wikipedia is not a place to give one's opinions unless they are shared by the wider community (for instance, "mathematics is important" is my opinion which is clearly shared by a large community; on the other hand, "Wikipedia will never succeed as an encyclopedia" is somewhat disputable). Thus, since there is not sufficient evidence supporting the idea that there are only two "best known" examples of Bézout domains, I do not see it as appropriate to include this comment in the article. --PST 09:32, 19 June 2009 (UTC)