Talk:Primary decomposition

Non-Noetherian rings?

If I'm not mistaken, the ideal $(X_{i}^{2}\mid i\in \mathbb {N} )\subset k[X_{i}\mid i\in \mathbb {N} ]$ has no primary decomposition because it has infinitely many associated primes, $(X_{i})$ for $i\in \mathbb {N}$.--gwaihir 23:28, 28 September 2006 (UTC)

We need to define what primary decomposition is, but that ideal does have one in general sense. WATARU 02:43, 30 September 2006 (UTC)

[KWR] Yes, this is a problem which keeps the current statement from being specific enough. The main points needing to be noted are that the primary decomposition of the ideal I has some finite number m of terms, and that the associated prime ideals are unique (up to permutations)---taking for granted that in writing I = \intersection_i=1^m Q_i, no Q_i contains the intersection of the other terms, so that the associated primes are distinct. Hence I have taken the liberty of flagging the theorem statement as incomplete.

It is also noteworthy that although the base field k should be algebraically closed for the "Ideal-Variety / Algebraic-Geometric Correspondence" to hold in full force, the Lasker-Noether Theorem holds over any base field k. My foremost source for all of this is pp206-209 of Cox, Little, and O'Shea, Ideals, Varieties, and Algorithms, 2nd ed., Springer-Verlag, 1996, which is searchable at Amazon <a HREF="http://www.amazon.com/Ideals-Varieties-Algorithms-Computational-Undergraduate/dp/0387946802/ref=si3_rdr_bb_product/104-7270086-9457506">here</a>.

Non-Noetherian rings are beyond my direct experience---I use polynomial ideals and computer algebra packages (notably <a HREF="http://www.singular.uni-kl.de/">Singular</a>) to investigate (algebraic) computational complexity theory, and basically all of them have routines that implement this theorem. In fact, I think having this theorem (with "finite") can be used as one of many equivalent conditions for being a Noetherian ring itself---?---or maybe not, MathReference.com calls such a ring <a HREF="http://www.mathreference.com/id-pry,intro.html">Laskerian</a>! The paper A. Seidenberg, "On the Lasker-Noether Decomposition Theorem", Amer. J. Math. 106:3 (June 1984), 611-638 appears to address the issue. Quick source checks do show near-but-not-full consensus on whether using the term "primary decomposition" entails "finite":

() Wikipedia does not have a separate page for primary decomposition---it forwards here. () PlanetMath says yes <a HREF="http://planetmath.org/encyclopedia/PrimaryDecomposition.html">here</a>. () Robert Ash of UIUC says yes <a HREF="http://www.math.uiuc.edu/~r-ash/ComAlg.html">here</a>, in a fairly general context. () So do SpringerLink <a HREF="http://eom.springer.de/P/p074450.htm">here</a> and MathReference.com <a HREF="http://www.mathreference.com/id-pry,decomp.html">here</a>. () But Wolfram MathWorld's entry <a HREF="http://mathworld.wolfram.com/PrimaryIdeal.html">here</a> does not have "finite". (I thought I had another reference, now can't find it.)

KWRegan 02:04, 8 January 2007 (UTC)

[KWR] Edited again since indentation caused funny boxing/spacing, so [KWR] marks my start.

KWRegan 02:06, 8 January 2007 (UTC)

[KWR] The change that was made decides to include "finite" in the definition of "primary decomposition", and that is fine by me.

69.204.22.43 (talk) 03:54, 28 February 2008 (UTC)

Comment

Tag this article as a 'technical article.' —Preceding unsigned comment added by 71.111.251.229 (talk) 02:28, 15 December 2007 (UTC)

fundamental theorem of finitely generated abelian groups

The fundamental theorem of finitely generated Abelian groups is generalized by the modular extension of the Lasker–Noether theorem, not by the theorem itself, for Abelian groups are modules over Z. —Preceding unsigned comment added by 85.146.200.37 (talk) 21:43, 20 May 2010 (UTC)

Move to primary decomposition

The following discussion is an archived discussion of a requested move. Please do not modify it. Subsequent comments should be made in a new section on the talk page. Editors desiring to contest the closing decision should consider a move review. No further edits should be made to this section.

The result of the move request was: move. -- tariqabjotu 02:06, 24 August 2013 (UTC)

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Lasker–Noether theorem → Primary decomposition – My impression is that the new name is more commonly used. I also don't know why the current name is preferred to the new name. (e.g., I don't see any discussion here.) Relisted. BDD (talk) 16:25, 15 August 2013 (UTC) Taku (talk) 01:16, 6 August 2013 (UTC)

• Support: I agree that "primary decomposition" is more commonly used than "Lasker-Noether theorem". The reason is probably that the theorem, an existence theorem, is rarely explicitly used. Commonly, people use it implicitly to work with the primary decomposition, with sentences like "Let us consider a primary decomposition, ...". The name of the existence theorem is not easy to insert in such a sentence. A consequence is that many young researchers that commonly use primary decomposition do not even know the name of the theorem. D.Lazard (talk) 09:19, 6 August 2013 (UTC)
• Comment: Two things. Firstly, I don't understand why anybody would use these words interchangably. "Primary decomposition" is not a theorem, it is a statement about structure of a submodule. Primary decompositions can exist in rings which aren't Lasker rings. That makes the objection about "which term is preferred" a non-objection. (But if anyone is still not convinced that the label is widespread, using google books it's easy to find the Noether-Lasker theorem called by name by Jacobson, Macaulay, Eisenbud, Lam, Rotman, Isaacs, Dauns, Passman, Knapp, Grillet, Jain and then lots of other authors I can't vouch for.) Secondly, I think a good place for the Noether-Lasker theorem is in a section in an article entitled "primary decomposition." So I support renaming the article and shifting focus to what a primary decomposition is, and I oppose using the phrases "primary decomposition" and "Noether-Lasker theorem" synonymously or insinuating that one of the terms is obsolete. Both terms are necessary. Rschwieb (talk) 14:06, 6 August 2013 (UTC)

  Of course, the primary decomposition is not the name of the theorem :) But the two terms do get used interchangeably: for instance, associated prime is using the phrase Lasker-Noether primary decomposition. It is also true that some authors do use only "primary decomposition" and never use "Noether-Lasker theorem" (e.g., Matsumura and if I remember correctly Atiyah-Macdonald). Whether that's correct, historically or otherwise, is beyond our interests (us as Wikipedia editors). Anyway, I can agree not to "purge" Noether-Lasker theorem entirely from Wikipedia: this request is not about that. -- Taku (talk) 15:40, 8 August 2013 (UTC)

  Ah well sure, if someone asked me if I preferred "Noether-Lasker decomposition of a module" or "primary decomposition of a module" I would have wanted the latter :) Rschwieb (talk) 17:14, 8 August 2013 (UTC)

The above discussion is preserved as an archive of a requested move. Please do not modify it. Subsequent comments should be made in a new section on this talk page or in a move review. No further edits should be made to this section.

First primary decomposition algorithm

By this old edit, R.e.b. credited Grete Hermann for the first primary decomposition algorithm. As far as I remember (I have not the article under hand), Hermann's article does not contains any such algorithm: its main focus is about linear algebra in polynomial rings (ideal membership, sygyzies, ...), and its consequences. Primary decomposition is not a direct consequence of linear algebra, partly because it requires factorization. Thus Hermann's article must be checked on this point. As far as I know, the first primary decomposition algorithm in polynomial rings over fields (for which pth roots are computable, where p is the characteristic) appears in Seidenberg's article Constructions in algebra (1974). For polynomial rings over the integers, an algorithm appears in another Seidenberg's article of the same year. D.Lazard (talk) 09:31, 3 June 2016 (UTC)

New example "Embedded prime"

The following example has been recently added to the article:

Consider the scheme

$X={\text{Spec}}\left({\frac {\mathbb {C} [x,y,z]}{((y^{2}-x)^{3},z)\cap (z)}}\right)$

which is a copy of the $xy$-plane with a fattened quadric curve given by $y^{2}-x$. Then, $(y^{2}-x)$ is an embedded prime ideal since there is a non-trivial scheme theoretic structure along this curve in the $xy$-plane.

This example is wrong, as the ideal $((y^{2}-x)^{3},z)\cap (z)$ is simply the ideal (z). Moreover, an hypersurface can never be an embedded component, since an embedded component must have a smaller dimension that the considered scheme.

Moreover this example uses a terminology which is not defined nor used elsewhere in the article (this article is about ideals not schemes.

For these two reasons, I'll revert this edit. D.Lazard (talk) 09:33, 20 September 2017 (UTC)

Hi D.Lazard, thanks for pointing this out. I should have written the ideal $((y^{2}-x)^{3}z,z^{2})$. This will have the embedded prime $(y^{2}-x,z)$.

Requested move 25 April 2020

The following is a closed discussion of a requested move. Please do not modify it. Subsequent comments should be made in a new section on the talk page. Editors desiring to contest the closing decision should consider a move review after discussing it on the closer's talk page. No further edits should be made to this discussion.

The result of the move request was: Not moved—a merge request is the right way to do this buidhe 20:48, 2 May 2020 (UTC)

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Primary decomposition → Primary decomposition and associated prime – In textbooks on commutative algebra, primary decomposition and associated primes are almost never discussed separately. Thus, maintaining separate articles for primary decomposition and associated primes is a bit weird. I suggest we move the article to this new name and move the materials in "associated prime" that concern the commutative case to this article. Taku (talk) 03:20, 25 April 2020 (UTC)

• Oppose. Such a move makes no sense before merging the articles. For the moment, I have no clear opinion about such a merge. The reason is that both articles give a WP:UNDUE weight to the noncommutative and module cases. As far as I know, most applications of the concepts use only the case of ideals in a commutative Noetherian ring. So most readers may be confused by the introduction at the beginning of the article of technical concepts that they do not need. A witness of this biased organization of the article is that despite most of the content is about the commutative case some important facts are lacking, such as the well behavior under localization, and the fact that the isolated primary components may be obtained as the inverse image of the localization of the ideal by a well chosen element. So, the articles should be rewritten (and possibly merged) for putting the commutative case of ideals at the beginning, and then explaining what must be changed for generalizing to other cases. D.Lazard (talk) 07:45, 2 May 2020 (UTC)

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The discussion above is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.

@D.Lazard: I'm sorry for late response. I think we both agree that the current organization of the materials is not good; i.e., problematic emphasis on module or non-commutative cases. We disagree on specific resolutions to the problem. In any case, I have started the draft article Draft:Primary decomposition and associated prime which is meant to replace the current article as well as the commutative portion of associated prime. I have also begun implementing some of your suggestions like starting the article with the ideal case first. It is not finished but you can see the type of the article I have in mind. What do you think? -- Taku (talk) 05:35, 23 May 2020 (UTC)

Since the draft is now more or less ready to be published, if there is no objection from you or other editors, I am planning to replace the current article with the draft in the next few days. (No need to rush but also no need to wait if there is no issue). -- Taku (talk) 23:23, 26 May 2020 (UTC)

The new version has been published. —- Taku (talk) 23:35, 27 August 2020 (UTC)

"Coprimary" listed at Redirects for discussion

The redirect Coprimary has been listed at redirects for discussion to determine whether its use and function meets the redirect guidelines. Readers of this page are welcome to comment on this redirect at Wikipedia:Redirects for discussion/Log/2023 May 7 § Coprimary until a consensus is reached. 1234qwer1234qwer4 13:50, 7 May 2023 (UTC)