The $p$-adic valuation ring

Every nonzero rational number can be written uniquely as $x=u{p}^n$ with $n\in \mathbb{Z}$ and $u=a\mathrm{/}b\in {\mathbb{Q}}^{\times }$ having both $a$ and $b$ prime to $p$: factor all powers of $p$ from the numerator and denominator and collect the difference in the exponent $n$. Define $v(0)=\mathrm{\infty }$ and $v(x)=n$ for $x\mathrm{\ne }0$. Then $v(xy)=v(x)+v(y)$, the ultrametric inequality follows by factoring out the smaller power of $p$, and $v$ is surjective because $v(p)=1$. Its nonnegative locus is exactly ${\mathbb{Z}}_{(p)}$, so [F1] makes ${\mathbb{Z}}_{(p)}$ a DVR with uniformiser $p$.

By [L1], the units are exactly the elements of valuation $0$, namely the fractions with numerator and denominator both prime to $p$. By [L2], every nonzero ideal is generated by $p^n$ for a unique $n\ge 0$.