ring of $S$-integers

Definition.

Let $K$ be a number field and let $S$ be a finite set of absolute values of $K$, containing all archimedean valuations. The ring of $S$-integers of $K$, usually denoted by $R_S$, is the ring:

$$R_S=\{k\in K:\nu (k)\ge 0\mathit{\text{ for all valuations }}\nu \notin S\}.$$

Notice that, for any set $S$ as above, the ring of integers of $K$, ${𝒪}_K$, is always contained in $R_S$.

Example.

Let $K=\mathbb{Q}$ and let $S=\{{\nu }_p,|\cdot |\}$ where $p$ is a prime and ${\nu }_p$ is the usual $p$-adic valuation, and $|\cdot |$ is the usual absolute value. Then

$$R_S=\mathbb{Z}\left[\frac{1}{p}\right]$$

, i.e. $R_S$ is the result of adjoining (as a new ring element) $1/p$ to $\mathbb{Z}$ (i.e. we allow to invert $p$).