I don't know whether this question is a bit too vague for MO or not, so feel free to delete it if you see fit.
The p-adic integer is defined by taking the inverse limit $\ldots \mathbb{Z} / p^2 \rightarrow \mathbb{Z}/p $.One way to see the p-adic integers is to see it as dealing with $ \mathbb{Z} / p, \mathbb{Z} / p^2, \ldots $ at the same time. So $p$-adic integers allow us to see the structure of the ring of integers at the prime $p$. Taking the fractional field we obtain the $ p$-adic rational field $\mathbb{Q}_p$.
This construction is useful in study the arithmetic of the ring (field). For example, in the theory of class field theory, we study the question in $\mathbb{Q}_p$ first and glue them together and do something more to get the solution for $\mathbb{Q}$.
I want to ask why it is not possible for us to construct a structure that will allow us to see the ring of integer at two primes $p,q$ together and see how do they interacts? The analog of the above inverse limit construction seems to still work though not an intergral domain. However, we can still localize where it is possible.
Here is the motivation for the above question. We know that the global question are not solve by simply gluing together solution for local question. I attribute that to the fact that the primes does not play alone but interact with one another. An illustration of this can be seen through the fact that the product of all normalized absolute value is 1. So my question is why not isolate two primes to understand how they are interacting with one another instead of looking at all of them at once. I think there is some complicated issue that will arise from this. Just want to know what they are.
A more particular question may be like this: Let call the construction obtained above $\mathbb{Q}_{p,q} $. Is there something in the same vein of class field theory for this object. What is the obstacle in having such a theory. I am vaguely know that we have a more general Galois theory not only for fields but for rings also.
Best Answer
As stankewicz said, it is a general principle in number theory that whenever only finitely many primes are involved, they act "independently" in the sense that analyzing what is happening locally at each prime separately is enough to understand what all the finitely many primes are doing. One example of this is the Chinese Remainder Theorem.
Here is another: if you want a version of the integers with two primes $p$ and $q$, start with $\mathbb{Z}$ and invert every prime $\ell \not \in \{p,q\}$. This gives a semilocal ring with $(p)$ and $(q)$ as the nonzero primes. Similarly, you can do this with any two prime ideals $\mathfrak{p}, \mathfrak{q}$ in a Dedekind domain $R$. But the ring you get is not very interesting: it is a semi-local Dedekind domain, hence its class group is trivial, very likely the unit group $R^{\times}$ will be infinitely generated, etc. This domain is the intersection of the two DVRs $R_{\mathfrak{p}}$, $R_{\mathfrak{q}}$, and everything you want to know about it can be reduced to the DVRs. The same with two replaced by any finite set...