# 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:

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

Notice that, for any set $S$ as above, the ring of integers of $K$, $\mathcal{O}_{K}$, 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 $p$).

Title ring of $S$-integers RingOfSintegers 2013-03-22 15:57:27 2013-03-22 15:57:27 alozano (2414) alozano (2414) 4 alozano (2414) Definition msc 13B22 ring of S-integers