Let $R_1$ and $R_2$ be two DVRs inside a field $K$.
Show that $R_1 \cap R_2$ is a semilocal PID.
I know that a DVR is a local PID. So my guess is the maximal ideals of $S = R_1 \cap R_2$ are $S \cap m_i$ where $m_i$ is the unique maximal ideal in $R_i$.
How do I finish the problem and also how to show it's a PID?
See Matsumura, Commutative Ring Theory, Theorem 12.2. It says the following
Let $K$ be a field and $R_1,\dots,R_n$ valuation rings of $K$ with $\mathfrak m_i$ the maximal ideal of $R_i$ and $R_i \not\subseteq R_j$ for $j\neq i$. Define $A=\bigcap_{i=1}^n R_i$. Then $A$ is semilocal with maximal ideals $\mathfrak p_i=\mathfrak m_i\cap A$. Moreover $A_{\mathfrak p_i}=R_i$. If each $R_i$ is a DVR, then $A$ is a PID.
