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I remember reading somewhere (most probably in Lang's Algebra) that integral domain in also known by the name "entire ring". I was thinking that is it somehow connected with complex analysis, but unfortunately I could not figure out much. I know that if $ \Omega$ is a domain in $\mathbb C$ then $R=\{f: \Omega \to \mathbb C: f \text { is holomorphic}\}$ is an integral domain.

Is it true that every integral domain can be obtained as ring of holomorphic function of some domain? Also what might be the possible reason for using the terminology 'entire ring' for integral domain?

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Arpit Kansal
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    See here. In line $7$ the exact reference of Lang is given, too. Most probably Lang took it from Bourbaki (entier=integral). – Dietrich Burde Oct 03 '16 at 17:16
  • "Is it true that every integral domain can be obtained as ring of holomorphic function of some domain?" No, as it would imply an absolute upper bound for the cardinality of an integral domain. – quid Oct 03 '16 at 17:16
  • for what it's worth: in French - "entier" means whole, which has an integral feel. edit - I see @DietrichBurde has made the point, but I'll leave mine to represent la francophonie – peter a g Oct 03 '16 at 17:16
  • @quid: How do you get bound on the cardinality of space of holomorphic functions of a domain? – Dontknowanything Oct 03 '16 at 17:21
  • This is the real meaning of Entire Ring. – Dietrich Burde Oct 03 '16 at 17:22
  • @DietrichBurde - so much for la francophonie.... On the other hand, we have "le spectre de l'anneau" which is way better than the spectrum of the ring – peter a g Oct 03 '16 at 17:23
  • Yes, I agree. In French "entire ring" then is L'Anneau des Nibelungen - hmm. – Dietrich Burde Oct 03 '16 at 17:27
  • @peterag but in French the notion is called "anneau intègre." – quid Oct 03 '16 at 17:29
  • @Dontknowanything certainly there cannot be more such functions than functions from the complex number to the complex numbers. (One could fo better than this, but that seems besides the point.) – quid Oct 03 '16 at 17:33
  • @quid: Are you saying that cardinality of $H(\Omega)$ is less than cardinality of $H( \mathbb C)$? ($H(\Omega)$ is ring of Homolomorphic function).Why so? – Dontknowanything Oct 03 '16 at 17:40
  • @quid - It's possible that there were competing words in the beginning? I wonder whether, actually, the entire notion (haha) came from German mathematics first? I doubt any one says "ein Ring ist ganz." In any case "whole" and "entire" are synonyms - but so is "integral." Maybe "intégral" would have been a better choice than "intègre" - and integral would have been better than entire? – peter a g Oct 03 '16 at 17:43
  • @Dontknowanything no, I am not saying this (although it would be true, with non-strict inequality). I wrote that the cardinality in question is at most the cardinality of the set of "functions from the complex number to the complex numbers." There is no mention of holomorphy there. This is trivial true by extending by $0$ to the whole complex plane. – quid Oct 03 '16 at 17:44
  • @DietrichBurde - please see my previous comment: what is "entire ring" in German? – peter a g Oct 03 '16 at 17:44
  • @quid: I see,i dint read your comment properly."Although it would be true with non strict inequality",how? – Dontknowanything Oct 03 '16 at 17:48
  • @peterag Actually one might say a "ein Ring ist ganz" but it'd mean something else, namely that it is integrally closed. The particular usage would be somewhat unusual. But to say "ganzer Ring" in that sense is not that unusual. – quid Oct 03 '16 at 17:49
  • @quid in the name of full disclosure: I speak French at home, but learnt math mostly in English - so my initial comment was purely to suggest possible etymology, not usage. – peter a g Oct 03 '16 at 17:53
  • @quid so what is an entire ring in German? – peter a g Oct 03 '16 at 17:55
  • @Dontknowanything by continuity it suffices to know a holomorphic function on a dense set; there is a countable dense set so $H(\Omega)$ is at most $|\mathbb{C}^{\mathbb{N}}|$ which is just $|\mathbb{C}|$. – quid Oct 03 '16 at 17:56
  • @peterag what precisely do you want to know? If you translate it litteraly it'd be "ganzer Ring" which as I already explained would be understood as "integrally closed ring." In the meaning as in OP it would be "Integritätsbereich" or also "Integritätsring" (though more rare I think). The matching adjective would be "integer" but it might not be used that much in that context (though sometimes it is). – quid Oct 03 '16 at 18:04
  • @peterag Entire Ring, or integral ring is "Integritätsring" in German. On the other hand, the ring of integers in an algebraic number field is "Ganzheitsring". – Dietrich Burde Oct 03 '16 at 18:09
  • @quid - thanks... and for what it's worth, though I suppose you know, "Bereich" = "domain" – peter a g Oct 03 '16 at 18:24
  • @DietrichBurde - thank you! – peter a g Oct 03 '16 at 18:24

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Is it true that every integral domain can be obtained as ring of holomorphic function of some domain?

No, this is not true. For one thing, this would imply an absolute upper bound on the cardinality of an integral domain.

Moreover, this connection is not really the historical reason for the name, see where does the term "integral domain" come from?

Also what might be the possible reason for using the terminology 'entire ring' for integral domain?

I do not know what the actual reasoning was, but a a reason might be that "integral" is used in a different sense in ring theory, too, namely an element is called integral over a ring $R$ if it is the root of a monic polynomial over $R$; and, a domain is called integrally closed if it contains all the integral elements from its quotient-field.

Both in French and in German two distinct words are used to signify those two notions, and one might want to follow the same practice in English.

Namely "intègre" (F) and "integer" (G) for "integral" as in "integral domain" and "entier" (F) and "ganz" (G) for "integral" as in "intrgal element."

What is strange though is that if this would be adopted the English usage would be somehow just the other way round relative to the French and German one, in that "entire" would not correspond to "entier" and "ganz."

It might be further worth noting that "intègre" (F) and "integer" (G) rather evoke the meaning "integrous," which would not be completely non-intuitive either (though it is not the historical motivation in German).

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  • Maybe I should add that beyond the adjectives, in French usage of "ring" ("anneau") is common, while in German "domain" ("Bereich") is common. Indeed, the English term derives from the German one. – quid Oct 03 '16 at 20:11