I have a basic question about rings of fractions.

Let $R$ be a commutative integral domain with quotient field $K$, $\mathfrak p$ a non-zero prime ideal of $R$ and $R_{\mathfrak p}$ the localization of $R$ at $\mathfrak p$.

Is it true that $K$ is also the quotient field of $R_{\mathfrak p}$? If so how I show this?

Thanking you in advance.

  • 3
    $\begingroup$ If by “quotient field” you mean the total ring of fractions, it is true, and you show it simply by writing down what the elements of each turn out to be. $\endgroup$ – Lubin Apr 3 '14 at 21:24
  • $\begingroup$ It might be useful that the fraction field the localization at the zero prime ideal. $\endgroup$ – user2345215 Apr 3 '14 at 21:45

Hint $\ $ The universal mapping property of localizations (or fraction fields) yields an easy test for isomorphism, see the Corollary below, from Atiyah & MacDonald, Commutative Algebra, p. 39.

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