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[parent] ring of $S$-integers (Definition)
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:
$\displaystyle R_S=\{ k\in K : \nu(k)\geq 0$    for all valuations $\displaystyle \nu \notin S \}.$

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

Example 1   Let $ K=\mathbb{Q}$ and let $ S=\{\nu_p,\vert\cdot\vert\}$ where $ p$ is a prime and $ \nu_p$ is the usual $ p$-adic valuation, and $ \vert\cdot\vert$ is the usual absolute value. Then
$\displaystyle R_S=\mathbb{Z}\left[\frac{1}{p}\right]$
, 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$).



"ring of $S$-integers" is owned by alozano.
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Other names:  ring of S-integers

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Cross-references: prime, contained, ring of integers, ring, valuations, archimedean, absolute values, finite set, number field

This is version 1 of ring of $S$-integers, born on 2006-06-07.
Object id is 7970, canonical name is RingOfSIntegers.
Accessed 1086 times total.

Classification:
AMS MSC: 13B22 (Commutative rings and algebras :: Ring extensions and related topics :: Integral closure of rings and ideals ; integrally closed rings, related rings )
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