What kind of arithmetic information does the ring of integers in an infinite extension carry? The fact that the ring of integers in a finite extension of $\Bbb Q$ is a Dedekind domain and purely algebraic properties of Dedekind domains are absolutely essential for algebraic number theory. So it seems natural to ask about infinite extensions next.
While the (not necessarily absolute) Galois groups of infinite algebraic extensions play a role e.g. in Iwasawa theory, it seems to me that the rings of integers of infinite algebraic extensions receive little attention. (Though this may just be a lack of knowledge on my part.)   
There is of course the classical result due to Dedekind that the ring of all algebraic integers is a Bezout domain. There are also notions in commutative algebra to describe what kind of ring the integral closure in an infinite algebraic extension will be. (The fact that one may call Prüfer domains "arithmetical domains" makes this question seem more pertinent.)
But while knowing for example the class group of the ring of integers in a finite extension can have striking consequences, it is not obvious to me what kind of arithmetic information we get for infinite algebraic extensions when we know for example class group of the ring of integers or that it is a Bezout domain. And I've never seen general results about Prüfer or Bezout domains applied in algebraic number theory.
So my question is the question in the title: What kind of arithmetic information can we get out of algebraic structure of the ring of integers in an algebraic extension? Do you know examples where results about Prüfer domains/Bezout domains/Krull domains etc. are applied to number theory?
 A: Picking up on the phrase "arithmetic information" in your question, let me give a brief answer coming from logic, although I recognize that this is likely not the answer for which you are looking. 
Logicians often measure the arithmetic information content of their mathematical structures by investigating which Turing degrees are encoded within those structures. To my way of thinking, anyone
taking the phrase "arithmetic information" seriously will end up talking about the Turing degrees.
For any nonstandard model $\mathcal{M}$ of Peano arithmetic PA, we
have the corresponding nonstandard ordered field of rational
numbers $\newcommand\Q{\mathbb{Q}}\Q^*$, which has the nonstandard
integers $\newcommand\Z{\mathbb{Z}}\Z^*$ sitting inside as a discretely ordered
ring, of which it is the quotient field. (But I'm not actually sure
whether $\Q^*$ can be algebraic over $\Q$.)
Logicians and researchers in the field known as models of PA
measure the arithmetic information content of such a structure by
means of the standard
system
of the model.
The standard system is the set of subsets
$A\subseteq\newcommand\N{\mathbb{N}}\N$ that are coded in the
model, in the sense that there is an element $n\in \N^*$, usually
nonstandard, such that $i\in A$ if and only if the $i^{th}$ prime
divides $n$ in $\N^*$. There are a huge variety of coding methods,
and all of them are equivalent for the purpose of defining the
standard system.
For example, a set $A$ is in the standard system of the model if
and only if there is a diophantine equation $p(x,\vec x)=0$ with
coefficients in $\Z^*$ such that $i\in A\iff p(i,\vec x)=0$ has a
solution in $\Z^*$, for $i\in\N$.
In countable models, a collection $S$ of sets $A\subseteq\N$ is the
standard system of a model of PA if and only if $S$ is a Boolean
algebra, closed under Turing reduction, and whenever $T\subset 2^{<\N}$ is an
infinite binary tree coded in $S$, then there is a branch
through $T$ in $S$.
The standard system of a nonstandard model of arithmetic measures
the arithmetic information content of the model because it
identifies upper bounds on which kind of arithmetic sets could be
defined in the model.
For example, some nonstandard models of arithmetic have their
standard systems consisting entirely of low
sets, which
have comparatively little information content; meanwhile, others
have a standard system closed under the Turing jump, and so their
information content exceeds the halting problem and much more.
A simple compactness argument shows that some nonstandard models of
PA have a standard system containing all sets of natural numbers.
It remains an open question, however, whether there is a Borel
definable such nonstandard model of PA, whose standard system
includes all sets $A\subset\N$.
