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)