Talk:Valuation ring

If V is a totally ordered group, a subgroup U of G is called an isolated subgroup of G if 0 ≤ y ≤ x and x ∈ U implies y ∈ U. The cardinality of the set of isolated subgroups of V is the height of V. If V is the trivial group, corresponding to the trivial valuation resulting from taking a field as a valuation ring, then V is of height zero.

Something is off with the counting, because the trivial group has one isolated subgroup, not zero. AxelBoldt 23:55, 10 May 2006 (UTC)

Fixed. Ringspectrum (talk) 19:50, 20 January 2009 (UTC)

Edit summary

In the edit summary, I meant to say "unless looking at a wrong edition, I cannot find it in the source." -- Taku (talk) 13:12, 17 July 2013 (UTC)

Maybe these will help with the terminology "refinement", "finer" and "coarser":

• Franz Viktor Kuhlmann; Salma Kuhlmann; Murray A. Marshall, eds. (2003). Valuation Theory and Its Applications 2. Fields Institute Communications. Vol. 33. American Mathematical Society. ISBN 0-8218-8590-1.
• Alexander Prestel; Charles N. Delzell (2001). Positive Polynomials: From Hilbert's 17th Problem to Real Algebra. Springer Monographs in Mathematics. Springer. p. 231. ISBN 3-540-41215-8.
• Murray Marshall (2008). Positive Polynomials and Sums of Squares. Mathematical surveys and monographs. Vol. 146. American Mathematical Society]]. p. 77. ISBN 0-8218-7527-2.

Additional. The assertion that a valuation ring is a maximal local ring with respect to dominance is Theorem I.6.1A of Robin Hartshorne (1977). Algebraic geometry. Graduate Texts in Mathematics. Vol. 52. Springer-Verlag. p. 40. ISBN 0-387-90244-9. Spectral sequence (talk) 16:46, 17 July 2013 (UTC)

Thank you very much for the references. I think I was thinking about maximal subrings and got confused; I just wanted to be sure it's correct. -- Taku (talk) 21:17, 17 July 2013 (UTC)

Construction section

I'm thinking of moving the construction section to valuation (algebra) since technicaly it is a construction of valuation and not a valuation ring. Does anyone object or otherwise has any opinion on the move? -- Taku (talk) 20:27, 9 August 2013 (UTC)

Typos: in this construction section Γ and G are both used to denote the value group.

Fixed. —- Taku (talk) 19:58, 12 April 2018 (UTC)

I deleted it because as far as I understand it it's just defining a Hahn series (and the Hahn series article mentions that you get the right valuation group. Can anybody let me know if I'm missing something here, or if there are any other concerns? Baum42 (talk) 23:58, 25 April 2021 (UTC)

@Baum42: I don’t think it’s s a good idea to just delete it. It’s ok that some materials of the article are duplicates of some other articles since otherwise the readers have to read several articles instead of one. —- Taku (talk) 05:29, 26 April 2021 (UTC)

@TakuyaMurata: To be clear, I didn't delete it because I mind the redundancy. My main issue is that it's just very confusing to have a top-level "Construction" section when it's not clear what's being constructed. When I read articles I often read the top-level sections out of order, so it took me a while to reconstruct what this section was trying to do, and only days later did I notice the "(see a section below)" in the previous section. One could ameliorate that by calling the section "Construction of a valuation ring with an arbitrary valuation group" or something like that, but that seems clunky. (We could also turn this section into a footnote to the sentence "Even further, given any totally ordered abelian group Γ, there is a valuation ring D with value group Γ.")

Then, as you mentioned, I'm not sure why this fact belongs here and not in valuation (algebra).

Finally, if we add it back, I think this paragraph should link to Hahn series as opposed to Formal power series but that should be trivial to do. Baum42 (talk) 07:07, 26 April 2021 (UTC)

@Baum42: This article is titled "valuation ring" so it seems clear that construction means construction of a valuation ring but yes adding the introductory sentence like "in this section, we construct a valuation ring out of a totally ordered abelian group" or something is needed to avoid confusion like one you got. I have completely forgotten that I made a proposal to move the materials but, since that section gives a good example of how to construct a valuation ring (really constructing a valuation), now I think it can belong here. —- Taku (talk) 03:34, 27 April 2021 (UTC)

Unclear language

In the Definition section, the line

 For a subring D of its field of fractions K the following are equivalent:

is awkward. My impression is they are trying to say what's more clearly phrased as

 Let D be an integral domain and K its field of fractions...

otherwise it's not clear what "its" is referring to. But I'm not confident enough that that's the intent to change it. M.C. ArZeCh (talk) 23:47, 15 April 2019 (UTC)

I agree and I'm not sure why this awkward wording is used here. ("field of fractions" suggests D is assumed to be an integral domain). I have there reworded it. -- Taku (talk) 00:36, 16 April 2019 (UTC)