Theorem 10.2 in Matsumura's commutative ring theory

Question

I'm reading "Commutative ring theory, Matsumura, but I don't understand the proof of Theorem 10.2 in Chapter 4. So, please ask you my question.

The statement is the following:

Let $K$ be a field, $A\subset K$ a subring, and $p$ a prime ideal of $A$. Then there exists a valuation ring $R$ of $K$ satisfying $R\supset A$ and $m_R\cap A=p$.

To prove this, we replace $A$ by $A_p$. My questions is:

(*) Why is it enough to show when $A$ is local and $p$ is the maximal ideal?

Thank you.

Answer 1

Imagine that we are able to do this for $A_p$; that is, there is a valuation ring $R$ of $\operatorname{Frac}A_p$ such that $A_p\subseteq R$ and $m_R\cap A_p = pA_p$. But $\operatorname{Frac}A_p = K$, $A\subseteq A_p$, and $pA_p\cap A = p$. So the same $R$ does the trick.