I have a question on localizations of Dedekind rings which I am learning about in an undergraduate class. Let $R$ be a Dedekind ring with quotient field $K$, $\mathfrak p$ a nonzero prime ideal in $R$. Let $R_\mathfrak p$ be the localization of $R$ at $\mathfrak p$.
One assignment question is to show that if $x\in K-R_\mathfrak p$ then $x^{-1}\in R_\mathfrak p$.
If $x=r/s\in K-R_\mathfrak p$ then $r\in R$ and $s\in \mathfrak p$. I don't see how we can deduce $r\notin \mathfrak p$ since we want $s/r\in R_\mathfrak p$.
A hint will be very useful. Thanking you in advance.
Best Answer
This exercise is asking you to prove that $R_\mathfrak{p}$ is a valuation ring (see the first definition in the Wikipedia entry: http://en.wikipedia.org/wiki/Valuation_ring#Definitions).
My answer to the following question explains this:
Discrete valuation ring associated with a prime ideal of a Dedekind domain