Valuation ring of a valued field

Definition

For a valued field $(K,v)$ , its valuation ring is

\mathcal O_v=\{x\in K:v(x)\geq0\}.

\mathcal O_v^\times=\{x\in K:v(x)=0\}, \qquad \mathfrak m_v=\{x\in K:v(x)>0\}.

Consequently $\mathcal O_v$ is a local ring, and $\mathcal O_v/\mathfrak m_v$ is its residue field.

Characterizing property

For every $x\in K^\times$ , totality of the value-group order gives

x\in\mathcal O_v\quad\text{or}\quad x^{-1}\in\mathcal O_v.

Conversely, a subring $R\subseteq K$ with this property is a valuation ring of $K$ for a suitable ordered value group.

References

Irving Kaplansky, “Maximal fields with valuations,” Duke Mathematical Journal 9 (1942), 303–321. DOI.