# 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 $K$, ${\mathcal{O}}_{K}$, 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 |
---|---|

Canonical name | RingOfSintegers |

Date of creation | 2013-03-22 15:57:27 |

Last modified on | 2013-03-22 15:57:27 |

Owner | alozano (2414) |

Last modified by | alozano (2414) |

Numerical id | 4 |

Author | alozano (2414) |

Entry type | Definition |

Classification | msc 13B22 |

Synonym | ring of S-integers |