Talk:Integral element

message: 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)
thread_title: Hello

message: 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)
thread_title: Title of article

message: 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)
replies: text: 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)
thread_title:  Title of Article 

message: 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)
replies: text: No, you don't need "reduced". I rewrote the article and this should be clearer now. -- Taku (talk) 16:36, 16 August 2011 (UTC)
thread_title:  Integral closure of a ring 

message: What does it mean? -- Taku (talk) 01:26, 14 March 2009 (UTC)
thread_title:  Completely integrally closed 

message: 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)
thread_title: Integral closure of an ideal 

message: 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)
thread_title: move to integral element

message: 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)
thread_title:  Mistake? 

message: 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)
thread_title:  Yet another problem with integral closure 

message: *The roots of unity and nilpotent elements are integral over Z.
Are not the roots of unity and the nilpotent elements in C the same?  If so, this example should read:
*The roots of unity (nilpotent elements) in C are integral over Z.
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)
replies: text: 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)
thread_title: Examples

message: 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)
replies: text: Yes, the wording can be improved, which I just did. -- Taku (talk) 10:10, 26 August 2014 (UTC)
thread_title:  Looks as if some wording should be improved 

message: Substituted at 18:23, 17 July 2016 (UTC)
thread_title: Assessment comment

message: An editor has identified a potential problem with the redirect :Integrally closed ring and has thus listed it for discussion. This discussion will occur at Wikipedia:Redirects for discussion/Log/2022 November 12#Integrally closed ring until a consensus is reached, and readers of this page are welcome to contribute to the discussion.  1234qwer1234qwer4 17:31, 12 November 2022 (UTC)
thread_title:  "Integrally closed ring" listed at [[Wikipedia:Redirects for discussion|Redirects for discussion]]