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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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    $\begingroup$ See here. In line $7$ the exact reference of Lang is given, too. Most probably Lang took it from Bourbaki (entier=integral). $\endgroup$ Commented Oct 3, 2016 at 17:16
  • $\begingroup$ "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. $\endgroup$
    – quid
    Commented Oct 3, 2016 at 17:16
  • $\begingroup$ 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 $\endgroup$
    – peter a g
    Commented Oct 3, 2016 at 17:16
  • $\begingroup$ @quid: How do you get bound on the cardinality of space of holomorphic functions of a domain? $\endgroup$ Commented Oct 3, 2016 at 17:21
  • $\begingroup$ This is the real meaning of Entire Ring. $\endgroup$ Commented Oct 3, 2016 at 17:22

1 Answer 1

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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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  • $\begingroup$ 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. $\endgroup$
    – quid
    Commented Oct 3, 2016 at 20:11

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