Proposition 3

*Let $A$ be a principal ideal domain and $K$ its field of fractions. The valuation rings of $K$ containing $A$ and distinct from $K$ are the rings of the form $A_{Ap}$, where $p$ is an extremal element of $A$.

Clearly $A_{Ap}$ ($p$ extremal) is a valuation ring containing $A$ and distinct from $K$ (Proposition 2). Conversely, let $V$ be a valuation ring distinct from $K$ and containing $A$. As $V \neq K$, $m(V)$ contains an element $x \neq 0$; writing $x = a/b$ where $a \in A$ and $b \in A$, it is seen that $A \cap m(V)$ contains the non-zero element $a$. As $A \cap m(V)$ is prime, it is of the form $Ap$ where $p$ is extremal in $A$. Then $A_{Ap} \subset V, pA_{Ap} \subset m(V)$, so that $V$ dominates $A_{Ap}$; as $A_{Ap}$ is a valuation ring of $K$ (Proposition 2), $V = A_{Ap}$.