Talk:Lasker–Noether theorem

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

[edit] Non-Noetherian rings?

If I'm not mistaken, the ideal has no primary decomposition because it has infinitely many associated primes, (X_i) for .--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)