Talk:Ideal theory

Comments

should this line

The ideal theory in question had been based on elimination theory, but in line with David Hilbert's taste moved away from algorithmic methods.

be writen like:

The ideal theory in question had been based on elimination theory, but , in line with David Hilbert's taste , it moved away from algorithmic methods. ~~

— Preceding unsigned comment added by 155.198.157.118 (talk • contribs) 21:05, 9 July 2007 (UTC)

Redirect to ideal (ring theory)

To respond to the recent edit to redirect the page to "ideal (ring theory)” with the edit summary: "(1) ideal theory has never been restricted to the commutative case (2) most assertions of the lead are either WP:OR or controversial (3) the body is not specifically on the theory of ideals in commutative rings, and most of this theory is not there"

(1) I disagree. As far as I can tell from algebra textbooks, there is really no substantial "non-commutative ideal theory" so it is fine to limit the article to the commutative case.

(2) Let’s fix the lead then and ditto to (3). The article should have much more stuff; the redirect is not a solution.

I will therefore revert redirecting. —- Taku (talk) 18:28, 28 May 2022 (UTC)

Here are some WP articles about ideals in non-commutative rings or about noncommutative rings that are defined by properties of their ideals: Jacobson radical, Absolutely flat ring, Simple ring, Jacobson ring, Artinian ring, Noetherian ring (many noncommutative Artinian and Noetherian rings are important in various area of mathematics). So, restricting ideal theory to the commutative case is definitevely a misnomer.

Moreover, the sections that are targets of a redirect must be transformed in stand alone articles. I did that for System of parameters, but forgot to do the same for other sections. When this will be done, the article would become empty, as there is no clear relations between these sections, except that they talk of ideals and modules.

In any case, there is no ideal theory in this article, as the different sections are very weakly related. D.Lazard (talk) 19:58, 28 May 2022 (UTC)

While ideals surely play an important role in the non-commutative ring theory, I haven’t heard the term "ideal theory" is used to refer to the study of ideals in non-commutative rings. On the other hand, the term "ideal theory" does exist in the commutative ring theory (e.g., there is a book with that title).

"the sections that are targets of a redirect must be transformed in stand alone articles." But why? Why do they need to be standalone articles? As Wikipedia is not a dictionary, it is not necessary to have a standalone article for each terminology. For instance, a system of parameters seems not significant enough topic to be a standalone article instead of being a part of a larger one. Finally, not sure what you mean by "no ideal theory in this article"; surely reduction theory is a part of ideal theory. So do other stuff. —- Taku (talk) 06:16, 29 May 2022 (UTC)

You don’t claim there shouldn’t be an article titled "number theory" because that’s just a collection of individual results. The question is whether ideal theory that is the study of ideals in commutative rings exists or not. If so, we need an article for that. If the present article lacks some results (e.g., primary decomposition, ramification theory, relation to valuation theory, etc), they need to be added, not having the article on the topic at all. —- Taku (talk) 06:23, 29 May 2022 (UTC)

About stand alone articles: Three sections of this article are about well established concepts that can be described in stand alone articles. It is the general trends in Wikipedia, that in such cases, the concepts are described in stand alone articles. This is why I have transformed into stubs the redirects to these sections. Note also that two of these stubs are more related to the theory of local rings than to the theory of ideals; so their inclusion here was questionable.

I have left here the section headings, with a link to the new stubs, since it may be possible that there remain some incoming links to these sections.

Presently, the article becomes rather ridiculous. It is not only almost empty, but it does not mention the main results of ideal theory (Primary decomposition, and also Hilbert's basis theorem, Hilbert's syzygy theorem, Hilbert's Nullstellensatz, and many others). So, I do not see a better solution than a redirect to Ideal (ring theory). This will be coherent with Module theory that redirects to Module (mathematics). D.Lazard (talk) 14:02, 29 May 2022 (UTC)

@D.Lazard: We probably need a tie-breaker. Again the question seems to be whether the subject called "ideal theory" exists in much the same way we have ring theory, valuation theory, module theory, ideal class theory, etc. etc. As you can see, most of materials in ring theory (or commutative ring) already exist elsewhere in Wikipedia but those articles do exist. This is because articles of different granulation is possible; an article like this one can discuss topics without too much details. If your issue is the lack of materials, I can try to expand it in coming days. -- Taku (talk) 06:58, 31 May 2022 (UTC)