Valuation Ideal of P-adic Norm on Rationals

Theorem
Let $\norm {\,\cdot\,}_p$ be the p-adic norm on the rationals $\Q$ for some prime $p$.

The induced valuation ideal on $\struct {\Q,\norm {\,\cdot\,}_p}$ is the set:
 * $\mathcal P = p\Z_{(p)} = \set{ \dfrac a b \in \Q : p \nmid b, p \divides a}$