Definition 1   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: $$R_S=\{ k\in K : \nu(k)\geq 0 \text{ for all valuations } \nu \notin S \}.$$

Example 1   Let $K=\Rats$ 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 $$R_S=\Ints\left[\frac{1}{p}\right]$$ , i.e. $R_S$ is the result of adjoining (as a new ring element) $1/p$ to $\Ints$ (i.e. we allow to invert $p$ ).