4
$\begingroup$

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.

$\endgroup$
2
  • 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
3
$\begingroup$

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.

enter image description here enter image description here

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.