fields - What is the prime spectrum of a Cauchy series ring?

In this answer I will treat the case in which $|text{ }|$ is not discrete.

I first claim that $mathfrak m_0$ is not the restriction of any proper ideal in
$k^{infty}.$ Indeed, choose $x in k$ such that $0 < |x| < 1$. Then $(x^i)$
is an element of $mathfrak m_0$ which is invertible in $k^{infty}$ (with
inverse equal to $(x^{-i})$, and so $mathfrak m_0$ generates the unit ideal
of $k^{infty}$.

This doesn't contradict anything; the maximal ideals of $k^{infty}$ pull-back
to prime ideals in $mathcal C(k)$ which are simply not maximal (as often happens
with maps of rings).

Furthermore, this pull-back is injective.

To see this, we first introduce some notation; namely, we
let $mathfrak m_{mathcal U}$ denote the prime ideal of $k^{infty}$ corresponding to
the non-principal ultra-filter ${mathcal U}$,and recall that $mathfrak m_{mathcal U}$ is defined as follows: an element $(x_i)$ lies in $mathfrak m_{mathcal U}$ if and only
if ${i , | , x_i = 0}$ lies in in $mathcal U$.

Now suppose that $mathcal U_1$ and $mathcal U_2$ are two distinct non-principal ultra-filters. Let $A$ be a set lying in $mathcal U_1$, but not in $mathcal U_2$.
Then $A^c$, the complement of $A$, lies in $mathcal U_2$.
Choose $x in k$ such $0 < | x | < 1,$ and let $x_i = x^i$ if $i in A$ and
$x_i = 0$ if $i notin A$. Then $(x_i)$ is an element of $mathcal C(k)$,
in fact of $mathfrak m_0$, and it lies in $mathfrak m_{mathcal U_2}$
but not in $mathfrak m_{mathcal U_1}$.

Thus $mathfrak m_{mathcal U_1}$ and $mathfrak m_{mathcal U_2}$ have distinct pull-backs.

So the map
Spec $k^{infty} rightarrow $ Spec $mathcal C(k)$
is injective
and dominant (since it comes from an injective map of rings), but is not surjective.
Choosing the valuation $|text{ }|$ allows us to add to Spec $k^{infty}$ (which is the
Stone-Cech compactification of $mathbb Z_+$) an extra point dominating all the
other points at infinity (i.e. all the non-principal ultrafilters), because the valuation now gives
us a definitive way to compute limits (provided we begin with a Cauchy sequence).