Talk:Completion of a ring

Untitled

Sorry this page is such a mess right now. Obviously the beginning will have to be rewritten. Joeldl 13:16, 15 February 2007 (UTC)

Always a ring extension?

The name "completion" sort-of suggests that the completion ring will be an extension of the original ring. However, the article does not come straight out and say that, and I think this is not always true, based on the following counterexample:

Let R be ring of rational functions generated by x and y/xⁿ for all n over some field. Let I be the maximal ideal (x). Then $y\in I^{n}$ for all n. If I understand the definitions right, then y maps to 0 under the natural homomorphism $R\to {\hat {R}}_{I}$, which is therefore not an injection.

But it may be that I don't understand the definitions. What is right? –Henning Makholm (talk) 03:28, 17 December 2009 (UTC)

The article also currently says "The canonical map from the ring to its completion is an isomorphism if and only if the intersection of the powers of I consists only of zero element of the ring." Should "isomorphism" be "injection"? That would be a reasonable answer to that question. And is the kernel of that canonical map the intersection of all the Iⁿ? 76.174.41.114 (talk) 07:57, 1 September 2010 (UTC)

Good catch. Fixed. Arcfrk (talk) 06:31, 2 September 2010 (UTC)

Is the completion complete?

Well, it is, but that depends on the topology. Suppose we take a Ring R, and an ideal I. Then it is true that \hat{R} is complete in the I-adic topology as an R-module. However, \hat{R} need not be complete in the \hat{I} topology. Counterxamples exists in the case that I is not finitely generated. I think this should be cleared. 85.64.171.226 (talk) 07:19, 29 May 2010 (UTC)

I haven't seen counterexamples explicitly mentioned in Bourbaki, but if this is a non-Noetherian issue then it is largely esoteric. Arcfrk (talk) 09:13, 29 May 2010 (UTC)

"Complete" topological space

I've never heard this term, and as far as I can tell it does not appear anywhere else on Wikipedia. Completeness is a metric property, not a topological one, and I-adic topologies are not in general metrizable (even Hausdorff) in the first place. Unless we mean to define some notion of "pseudo-metric completion," the terminology in this article needs to be refined.

Consider the article on localization: the most common multiplicative system at which to localize is the complement of a prime, in which case the result is a local ring - hence the terminology. The situation here is similar: The most common case in which completion is deployed it to complete a Noetherian local ring (or just local ring, depending on your terminology) at its maximal ideal. In this case, we have an actual metric, and the resulting metric space is actually complete. This is where the term "completion" comes from, and (just as not every localization is local) not every completion is complete, contrary to the claim of the article. 68.61.30.23 (talk) 15:22, 8 January 2013 (UTC)

Incidentally, I want to note that it actually does seem to me like the best option is to use a notion of pesudo-metric completion: in this sense, it is in fact the case that all completions are complete. That ought to be specified, however, and terms "complete topological module" and "complete topological ring" ought to be scrapped, as they are not precise. 68.61.30.23 (talk) 15:25, 8 January 2013 (UTC)

I think "complete" here means it coincides with the completion. But this needs a reference; for now, I have added a tag requesting the clarification. —- Taku (talk) 18:06, 2 July 2022 (UTC)

Completion with respect to a maximal ideal is local

I have just read that the completion of a commutative ring with respect to a maximal ideal is a local ring. This seems like it definitely deserves to be included in the article. Joel Brennan (talk) 21:51, 18 June 2022 (UTC)

It’s so well-known that apparently Wikipedia editors didn’t bother to mention it :( I have just added this (with a reference). I think we should also mention "Noetherian" cannot be dropped (if I remember correctly). —- Taku (talk) 16:30, 21 June 2022 (UTC)

"Idealwise separated" listed at Redirects for discussion

The redirect Idealwise separated has been listed at redirects for discussion to determine whether its use and function meets the redirect guidelines. Readers of this page are welcome to comment on this redirect at Wikipedia:Redirects for discussion/Log/2024 March 18 § Idealwise separated until a consensus is reached. 1234qwer1234qwer4 03:04, 18 March 2024 (UTC)

"Complete ring" listed at Redirects for discussion

The redirect Complete ring has been listed at redirects for discussion to determine whether its use and function meets the redirect guidelines. Readers of this page are welcome to comment on this redirect at Wikipedia:Redirects for discussion/Log/2024 April 6 § Complete ring until a consensus is reached. 1234qwer1234qwer4 14:30, 6 April 2024 (UTC)