I'm reading Matsumura's proof on independence of valuation (p.87, "Commutative Ring Theory"), and I understand most of the proof, except the last bit about principal ideal ring, and if anyone could help me I would greatly appreciate it:
$K$ a field, $R_1$,...,$R_n\subset K$ DVRs of $K$ not containing one another, $A=\cap_{i}^{n}R_i$. Let $\mathfrak{m}_i\subset R_i$ be the maximal ideals and $\mathfrak{p}_i:=\mathfrak{m}_i\cap A$. I can see that $A$ is a semi-local ring with maximal ideals $\mathfrak{p}_1$,...,$\mathfrak{p}_n$, and that $A_{\mathfrak{p}_i}=R_i$. But I don't see why $A$ is a PID. Here is a passage from the book:
$\mathfrak{m}_i\neq\mathfrak{m}_i^2$, and hence $\mathfrak{p}_i\neq\mathfrak{p}_i^{(2)}$, where $\mathfrak{p}^{(2)}:=\mathfrak{p}^2A_{\mathfrak{p}}\cap A$. Thus there exists $x_i\in\mathfrak{p}_i$ such that $x_i\notin\mathfrak{p}_i^{(2)}$, and $x_i\notin\mathfrak{p}_j$ for $i\neq j$ (I understand up to here); then $\mathfrak{p}_i=x_iA$ (I don't understand this). If $I$ is any ideal of $A$ and $IR_i=x^{\nu_i}_iR_i$ for $i=1$,...,$n$ then it is easy to see that $I=x_{1}^{\nu_1}...x_{n}^{\nu_n}A$ (I don't see this).
I'm sorry for this rather pedantic question but I have no one to help me...