Quotient Field of a Localization – Abstract Algebra

Question

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.

Answer 1

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.