Now I try to do exercise 12.4 in the book "Commutative ring theory" by H. Matsumura.
Let $A$ be a Krull domain, $I\subseteq \mathfrak p$ and $\mathfrak p$ is a height $1$ prime ideal of $A$. I don't know how to prove the following statement:
$xI^{-1}\subseteq A_{\mathfrak p}$ is equivalent to $x\in IA_{\mathfrak p}$.
If $x\in IA_{\mathfrak p}$, then easily $xI^{-1}\subseteq A_{\mathfrak p}$. But how to prove the opposite direction of the statement above? Can someone explain it to me?
Thanks a lot!