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Is it true that every localization of an integral domain is isomorphic to a subring of its field of fractions? How are the localizations of an integral domain related to its field of fractions? Is there a handy criterion to tell whether a subring of an integral domain has the same field of fractions as the overring? I've read that $k[x^2,x^3]\subset k[x]$ ($k$ a field, algebraically closed if this matters) have the same field of fractions, how come? I am very curious about these things.

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    $\begingroup$ Did you mean to ask if every subring is a localization? $\endgroup$ – Gone Jan 7 '14 at 0:12
  • $\begingroup$ no, sorry if I was unclear (silly me), I am asking that suppose you have an integral domain $I$, and you take $S^{-1} I$ ($S$ multiplicative), is $S^{-1}I$ is isomorphic to a subring of $\mathrm{Frac}(I)$? But what would the answer to your interpretation of my question be? $\endgroup$ – user88576 Jan 7 '14 at 0:30
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    $\begingroup$ How do you understand localizations in a way that the answer to that simpler question is not obvious? $\endgroup$ – Gone Jan 7 '14 at 0:34
  • $\begingroup$ I see that your question was a trivial one, so why do you pose it? $\endgroup$ – user88576 Jan 7 '14 at 0:38
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    $\begingroup$ Huh? $\phantom{}$ $\endgroup$ – Gone Jan 7 '14 at 0:45
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Answering your questions:

  1. Yes. A localization of a domain $R$ with respect to some multiplicatively closed subset $S$ of $R$ is a subring of $K$ (the field of fractions of $R$) consisting of the elements $m/s$, with $m \in R$ and $s \in S$.
  2. If you take $S = R\setminus\{0\}$, the localization of $R$ with respect the set $S$ is the field of fractions $K$ of $R$.
  3. The field of fractions of a domain $R$ is the smallest (with relation to inclusion) field that contains $R$. So, two domains have the same field of fractions if the smallest fields that contains each one are the same.
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Let $D$ be a domain an let $K(D)$ be its fraction field. Let $S$ be a multiplicative subset of $D$.

  • Is $S^{-1}D$ a subring of $K(D)$? Yes, to prove it remembre that injectivity is a local property of morphism.(We obtain that EVERY localization of $D$ is a subring of $K(D)$)
  • Have $k[x^2,x^3]$ and $k[x]$ the same fraction field? Yes: We need just to show that $x \in K(k[x^2,x^3])$, but you can obtain $x$ as the class of $x^3/x^2$.
  • Are there some handly criterions to show that $A \subset D$ is such that $K(A)=K(D)$? Well,seriously I don't know but i think there are not general results. It works certainly if $A$ and $D$ have the same integral closure.
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