Ring of integers and valuations in number fields Let $K$ be an algebraic number field and $\mathcal{O}_K$ its ring of integers, containing exactly those elements that satisfy a monic polynomial equation with coefficients in $\mathbb{Z}$. I was told the following definition: given a finite set $S$ of places of $K$ containing all Archimedian places, one defines the subring of $K$ of $S-$integers
$$ \mathcal{O}_S := \{x \in K \ \vert \ \nu(x) \ge 0 \ \forall \ \nu \notin S\}.$$
I was also told that it is always true that $\mathcal{O}_K \subseteq \mathcal{O}_S$. Therefore I wonder whether the equality
$$ \mathcal{O}_K = \{x \in K \ \vert \ \nu(x) \ge 0 \ \forall \ \nu \ \text{ultrametric}\}$$
is true for all number fields. It clearly holds for $\mathbb{Q}$.
A related question: one excludes Archimedian places because they don't (need to?) have an underlying valuation. Is there any deeper reason for this?
Also, do the prime ideals in $\mathcal{O}_S$ correspond $1:1$ to the ultrametric places in $S$?
 A: This is too long to be a comment, but there definitely is "deeper meaning" to the infinite places of a number field. In the analogous setting of function fields, the ring $\mathbb{F}_q[t]$ is our analogue of $\mathbb{Z}$, and $\mathbb{F}_q(t)$ is $\mathbb{Q}$, and in this setting, we have the direct geometric interpretation;  $\mathbb{F}_q(t)$ are the rational functions on the projective line $\mathbb{P^1}/\mathbb{F}_q$. In this setting, $\mathbb{F}_q[t]$ are the regular functions on the affine line, that is, the functions on $\mathbb{P^1}$ with their poles only at $\infty$. In this setting, the point/valuation at infinity is just like the others, and by considering all of them you get a much nicer theory.
So if we view $\mathbb{Z}$ as the functions on some extended version of $Spec(\mathbb{Z})$ with their only poles at "$\infty$", to get better, more complete results about number fields, we should consider the behaviour at "infinity", just as in the algebraic geometry case of curves/Riemann surfaces.
One nice example of this is to look at which elements of a number field are "globally defined" in that they have no poles, even at the infinite places. A reasonable interpretation of this is to ask for the elements $x\in K$ such that $|x|=1$ for all places, including the infinite ones. Its a nice exercise to show that this gives precisely the roots of unity in $K$, which gives some justification to the intuition of these as the field of constants of $O_K$.
This principle can also be seen in the adelic product formula for number fields (sum of zeros and poles of a meromoprhic function is 0), and determining the rank of the group of units in number fields (rank is number of points missing - 1).
