[Math] Localization of an integral domain and fields of fractions

abstract-algebracommutative-algebraring-theory

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.

Best Answer

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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