Talk:Krull dimension
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Some edits to improve the readability of the article
[edit]- No symbol has been specified for the Krull dimension of a ring, the definition should be changed to something like:
- We define the Krull dimension of R to be the supremum of the lengths of all chains of prime ideals in R and we denote it by (or simply when there is no risk of confusion).
- Incorrect use of punctuation: (geometers call it the ring of the normal cone of I.) should be changed to (geometers call it the ring of the normal cone of I).
- Add some links:
- the space of prime ideals of R equipped with the Zariski topology --> the space of prime ideals of R equipped with the Zariski topology
- The equality holds if R is finitely generated as algebra (for instance by the noether normalization lemma). ---> The equality holds if R is finitely generated as an algebra (for instance by the Noether normalization lemma).
- In the language of schemes, finitely generated modules are interpreted as coherent sheaves, or generalized finite rank vector bundles. ---> In the language of schemes, finitely generated modules are interpreted as coherent sheaves, or generalized finite rank vector bundles.
- Rename the Notes section as References;
- Introduce a Notes section for remarks and clarifications on the many facts listed in the article.
- Add the following note concerning the fact the height of is the Krull dimension of the localization of at to the Note section
This follows from the following observation: for any prime ideal consider the localization of to the multiplicative system which we denote by ; the natural map induces a bijection[ref 1]
defined by , with inverse .
- ^ Watkins, John (2007). Topics in Commutative Ring Theory. Princeton University Press. p. 64. ISBN 9780691127484.
Theorem 6.1
- I am concerned with the use of both I and I to denote an ideal: not only is this confusing for the reader, but the symbol I (or ) is also commonly used to denote either the set of imaginary numbers or the compact (especially in algebraic topology).
Please, let me know what you think.--Ale.rossi91 (talk) 23:08, 1 February 2020 (UTC)
Example
[edit]It seems to me that there is an error in computation of the Krull dimension of (Z/8Z)[x,y,z] : we get a chain of prime ideals of length four by adding the (0) ideal to the chain that is given : . Thus I think that the dimension is 4.
129.199.2.17 (talk) 11:38, 13 February 2009 (UTC)
- It's correct, because we don't count (0)? (Otherwise, the field would have the dimension 1.) -- Taku (talk) 21:25, 13 February 2009 (UTC)
- In fact, we are both wrong, and the article was correct. The ideal (0) is prime if and only if the ring is a domain. The example is not a domain, so (0) is not prime. In the case of a field, the only prime ideal is (0), because the whole ring (field) is never a prime ideal. Thus the dimension of a field is still 0.82.67.178.125 (talk) 22:33, 14 February 2009 (UTC)
Eh?
[edit]Mr. Billion 08:47, 12 Jan 2005 (UTC)
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