# Talk:Integral element

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Field: Algebra

I am the creator of this page and I am writing in response to the proposal by SatyrTN for speedy deletion. This is only a stub — it's meant to be expanded, and it was created literally hours ago. It's intended to have more than just the definition, eventually. I think anybody who knows some commutative algebra knows that this will be a useful topic to have done. Joeldl 09:59, 13 February 2007 (UTC)

## Title of article

I'd like input on this: Should the article be renamed to Integral extension? This does not directly include the idea of an integral element (without considering the ring generated by the base ring and that element), but "integral extension" is a more likely search phrase than "integrality". I suppose an alternative would be Integral element, or Integral dependence. Joeldl (talk) 05:29, 13 March 2008 (UTC)

The "Integrally Closed" page has a request for discussion regarding merging the "Integrally Closed" page into this page.

I vote for keeping them as separate pages. They are both useful in their own way, and, they really are two different concepts, even though they are within the same subject.

I think "if it ain't broke don't fix it" applies here. Also the merge operation has the possibility of "breakage" since the two pages contain different (and valuable IMHO) information.

It might be easier to maintain them as two separate pages. For example, the comment regarding Integrally Closed Groups in the Integrally Closed Talk section fits more natually with the context of separate pages.

I think that if the two pages are kept like they are, they should cross-reference each other at the "See also" section at the bottom because they have different information.

Keeping the two pages separate helps with the "search" problem mentioned in the previous note.

DeaconJohnFairfax (talk) 23:05, 30 June 2008 (UTC)

I vote for the merger. It looks like this page is already set up to easily incorporate everything on the integrally closed page regarding rings (in the normal schemes subsection for example). I think it would be unnatural, the way this article is heading, to not encompass all the information from the ring part of the integrally closed page. Then the only reason to keep the integrally closed page would be if it were extensive enough to merit a "See main article" link from the integrality page.
As for the content on ordered groups, this looks like it could be easily included as another paragraph in the article on ordered groups.
The integrally closed page could then become a disambiguation page. RobHar (talk) 23:32, 30 June 2008 (UTC)

Good Idea! DeaconJohnFairfax (talk) 16:44, 1 July 2008 (UTC)

## Title of Article

Regarding the title of this, I think "Integral Extensions" would be a better title than "Integrality." I like the plural (s) in the title because the page discusses Integral Extensions in a number of different contexts.

When I first saw the link to "Integrality" on the "Integrally Closed" page, I almost didn't go to it because I thought it would be talking about whether or not a rational number is an integer. In other words, I thought it might be an extremely elementary article.

Either "Integral extension" or Integral extensions" communicates the idea that this is advanced math, not elementary school (or grammar school) stuff. DeaconJohnFairfax (talk) 23:07, 30 June 2008 (UTC)

I think the idea is that this page discusses the notion of integrality to various degress of generality. Not everything discussed here is about integral extensions, some of it is about integral morphisms, or integrality of non-affine schemes. I think integrality is probably a more appropriate title. RobHar (talk) 23:42, 30 June 2008 (UTC)

I just noticed last night that Neukrick's new book on Class Field theory calls his section on Integrality, "Integrality." I also noticed that many Wikipedia math articles add a qualifier to specify what kind of Integrality they are talking about. I.e., this page does not talk about round (or integral) numbers versus numbers with fractional parts - like is done in elementary school). DeaconJohnFairfax (talk) 16:43, 1 July 2008 (UTC)

The fact that something is in a ring is a convention (functions from any set to the real numbers can be thought of as belonging to a ring of functions but you also can consider them without knowing about rings -- you could say such-and-such function is integral when talking about such-and-such type of functions without using the word ring).

So it would be wrong to change the title to 'integrality in rings.' 92.14.228.189 (talk) 07:21, 8 January 2011 (UTC)

## Integral closure of a ring

The article defines the integral closure of a reduced ring. Could someone with access to the reference add a note why/if reduced is required? 78.52.196.230 (talk) 11:02, 30 October 2008 (UTC)

No, you don't need "reduced". I rewrote the article and this should be clearer now. -- Taku (talk) 16:36, 16 August 2011 (UTC)

## Completely integrally closed

What does it mean? -- Taku (talk) 01:26, 14 March 2009 (UTC)

## Integral closure of an ideal

Is this definition right? The same definition but with I replaced by powers of I occurs in some papers and for elements of the ring is proven to be equivalent to the same equivalent formulation. 92.14.228.189 (talk) 09:50, 7 January 2011 (UTC)

Me again. I have corrected this. It was an error. Should be a_i \in I^i. 92.14.228.189 (talk) 07:23, 8 January 2011 (UTC)

Just to explain it a bit more: what Atiyah and Macdonald call 'integral closure' in case of no ring extension it is just the radical. But the earlier Zariski-Samuel, and all later texts and all articles, and indeed this Wikipedia article are not talking about the operation in Atiyah-Macdonald where a_i\in I is allowed rather than a_i\in I^i.

92.14.228.189 (talk) 07:27, 8 January 2011 (UTC)

## move to integral element

I moved the article to "integral element". Now that we have "integrally closed", this seems more appropriate. -- Taku (talk) 12:32, 14 August 2011 (UTC)

## Mistake?

Hi, I am new to studying this part of math, but I was wondering if there is a mistake on this page. In the beginning of the page, we write that if B consists only of elements that are integral over A, then we say B is integral over A. This makes sense, especially noting that B contains A. Later on in the page, we define a ring homomorphism f:A to B to be integral if f(A) is integral over B. This doesn't make sense to me since f(A) is a subring of B? I'm probably just confused. — Preceding unsigned comment added by 24.124.53.126 (talk) 23:03, 25 September 2011 (UTC)

## Yet another problem with integral closure

The phrase "integral closure" is used several times before it is defined (even casually). I suggest that at least a modest definition be given that precedes the first use of the phrase.Daqu (talk) 19:11, 4 February 2012 (UTC)

## Examples

Are not the roots of unity and the nilpotent elements in C the same? If so, this example should read:

Or was this example meant to include examples of ring extensions of Z other than (subrings of) C? If no one answers this question for me, I am inclined to make this change someday. Howard McCay (talk) 20:14, 18 August 2012 (UTC)

they don't have to be in the field of complex numbers. A root of unity is a root of 1; i.e., a solution of x^n = 1. For nilpotent, replace 1 by 0. Also, note any ring is a $\mathbb{Z}$-algebra. Thus, the example makes sense. (But, yes, it could be reworded for clarity.) -- Taku (talk) 20:50, 18 August 2012 (UTC)

## Looks as if some wording should be improved

One passage of the article begins as follows:

"If A, B are fields, then the notions of "integral over" and of an "integral extension" are precisely "algebraic over" and "algebraic extensions" in field theory (since the root of any polynomial is the root of a monic polynomial). The special case of greatest interest in number theory is that of complex numbers integral over Z; . . ."

OK, the case of the algebraic numbers being "integral over Z" may be isomorphic to the case of the algebraic numbers being integral over Q — which is a field. But it surely is not a "special case" of A and B being fields.

So, this is a very poor "special case", since it isn't one.Daqu (talk) 15:55, 22 August 2014 (UTC)

Yes, the wording can be improved, which I just did. -- Taku (talk) 10:10, 26 August 2014 (UTC)