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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
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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.
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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
471:
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 Knowledge article are not talking about the operation in Atiyah-Macdonald where
355:
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
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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
340:
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 Knowledge 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
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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.
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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.
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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.
229:
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.
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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.
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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.
513:
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.
226:
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.
189:(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
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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.
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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.
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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.
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Either "Integral extension" or
Integral extensions" communicates the idea that this is advanced math, not elementary school (or grammar school) stuff.
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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
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As for the content on ordered groups, this looks like it could be easily included as another paragraph in the article on ordered groups.
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767:, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.
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I moved the article to "integral element". Now that we have "integrally closed", this seems more appropriate. --
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until a consensus is reached, and readers of this page are welcome to contribute to the discussion.
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Keeping the two pages separate helps with the "search" problem mentioned in the previous note.
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No, you don't need "reduced". I rewrote the article and this should be clearer now. --
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Normal ring redirects here, but the article only talks mentions normal domains.
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Me again. I have corrected this. It was an error. Should be a_i \in I^i.
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Or was this example meant to include examples of ring extensions of
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So it would be wrong to change the title to 'integrality in rings.'
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The integrally closed page could then become a disambiguation page.
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An editor has identified a potential problem with the redirect
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So, this is a very poor "special case", since it isn't one.
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OK, the case of the algebraic numbers being "integral over
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to the case of the algebraic numbers being integral over
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I'd like input on this: Should the article be renamed to
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Yes, the wording can be improved, which I just did. --
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