Talk:Hilbert's basis theorem

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Surely the Hilbert Basis Theorem fails when there is an infinite number of variables. If this is the case, then the theorem needs to be specified a little more precisely.

Best Wishes, Max Murphy (Dr, Permutation Patterns).

Slightly more general?

immediately following is the case of a single variable, without multidegrees etc., it doesn't read well. 24.59.111.68 05:44, 15 November 2007 (UTC)

• the proposition "R Noetherian --> R[X] Noetherian" is a chainable proposition : If R is Noetherian then R[X] is Noetherian and so (R[X])[Y] is Noetherian and (R[X])[Y] is just R[X,Y] and so on. —Preceding unsigned comment added by Novwik (talk • contribs) 21:22, 29 December 2007 (UTC)

It's no left-ideal over R

In the second proof we read

Let b be the set of leading coefficients of members of a. This is obviously a left-ideal over R [...]

That's obviously wrong since 0 is not in b. 0 is never the leading coefficient of a polynomial. What is meant is the ideal generated by the leading coefficients. --Jobu0101 (talk) 17:41, 15 May 2014 (UTC)

Assessment comment

The comment(s) below were originally left at Talk:Hilbert's basis theorem/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

So called 'proof' needs to go. On the other hand, the various implications, including the history, have to be thoroughly explained. Arcfrk 04:22, 28 June 2007 (UTC)

Last edited at 04:22, 28 June 2007 (UTC). Substituted at 02:10, 5 May 2016 (UTC)

Poor language deserves course correction before it reaches the state of the "Titanic"

From the article: "Hilbert's basis theorem says that a polynomial ring over a Noetherian ring is Noetherian."

"a Noetherian ring is Noetherian"? This is what is constituted as knowledge, is this what mathematics becomes? Shakka, when the walls fell!

I understand the value of encoding abstract topics, and I understand the honor involved in having things named after you, but this simply serves to hold out anyone wanting to contribute to a meaningful conversation, when otherwise understandable terms are obfuscated behind language designed to keep out-groups ill-informed.

"Hilbert's basis theorem says that a polynomial ring over a ring that satisfies the ascending chain condition on left and right ideals has the property every increasing sequence of left (or right) ideals has a largest element; that is, there exists an n such that: I_n = I_(n+1)..." actually says something.

"Riemannian Manifolds with Riemannian curvature, is effectively Euclidean space with Euclidean triangles, where the Euler equation means that Newton's Law is an Approximation to Einstein's Equation" says absolutely nothing, unless you already studied history, not mathematics.

As mathematicians, "a Noetherian ring is Noetherian" should be unacceptable. 2601:646:9501:6D30:468D:F0E0:F980:E24F (talk) 18:54, 3 December 2023 (UTC)

"A Noetherian ring is Noetherian" is unacceptable, but the sentence is not in the article.

The article is mathematically correct, since all technical terms are linked to a an accurate definition and that these definitions are not circular.

Nevertheless, although mathematically correct, the article is far to be correct for an encyclopedia: Not only it lacks of the minimally needed explanations, which should include the definitions of fundamental technical words and phrases, but it lacks completely of context. In particular, the term "basis theorem" is not explained, it is not said that the theorem is, with Hilbert's Nullstellensatz and Hilbert's syzygy theorem, the first non constructive theorems, and that this revolutionized mathematics.

Fixing these issues requires some work, and and I can do this immediately. D.Lazard (talk) 09:20, 4 December 2023 (UTC)