Valuation ring

Remarks

Example: for a prime $p$, the ring $\Z_{(p)} = \set{a/b : a, b \in \Z,\ p \nmid b}$ is a valuation ring. Each nonzero rational number is $x = p^n a/b$ with $n \in \Z$ and $a, b$ prime to $p$; if $n \ge 0$ then $x \in \Z_{(p)}$, and if $n < 0$ then $x^{-1} = p^{-n} b/a \in \Z_{(p)}$. Non-example: $\Z$ is not a valuation ring, since neither $2/3$ nor $3/2$ is an integer. Every field is a valuation ring of itself. If $v$ is a valuation on a field $K$, then $\set{x \in K : v(x) \ge 0}$ is a valuation ring with fraction field $K$. Conversely, let $R$ be a valuation ring with fraction field $K$. Then $K^\times / R^\times$, ordered by $xR^\times \le yR^\times$ when $y/x \in R$, is a totally ordered group. The quotient map $K^\times \to K^\times/R^\times$ extends to a valuation whose ring is $R$. A valuation ring is a local ring, with maximal ideal the set of non-units, and any two of its ideals are comparable by inclusion.