I have been trying to study the prime ideals of the ring $R:=\mathbb Z [\{ \sqrt{p_n}\}_{n=1}^\infty]$, where $p_n$ denotes the $n$-th prime. This is how far I got: I could conclude, by means of the first isomorphism theorem and the linear independence of square roots of distinct square-free numbers over $\mathbb Q$, the following isomorphism $$\mathbb Z[\{x_n\}_{n=1}^\infty] / (x_n^2 - p_n : n \in \mathbb N) \cong R.$$ Indeed, we consider the map $\phi: \mathbb Z[\{x_n\}_{n=1}^\infty] \longrightarrow R$ acting as identity on $\mathbb Z$ and sending $x_n$ to $\sqrt{p_n}$ for each natural number $n$. Then $\phi$ is clearly a surjective ring homomorphism and has kernel equal to $I := (x_n^2 - p_n : n \in \mathbb N)$: that this ideal is part of the kernel is easily checked whereas for any $f$ in the kernel, there must exist a natural number $N$ for which $f \in \mathbb Z [x_1, \cdots , x_n]$, whence by inductively performing long division by the polynomials $x_1^2-2, \cdots , x_N^2-p_N$, I could write $f(x_1, \cdots , x_N)$ in the form $g(x_1, \cdots , x_N) + h(x_1, \cdots , x_N)$ where $g(x_1, \cdots , x_N)$ is an element of the ideal $(x_1^2 - p_1, \cdots , x_N^2-p_N) \subset I$ and $h (x_1, \cdots , x_N)$ is a (finite) integer linear combination of $1$ and squarefree monomials in the indeterminates $x_1, \cdots , x_N$. Consequently, we have $h(\sqrt 2, \cdots , \sqrt{p_n}) = 0$, which by the well-known result of $\mathbb Q$-linear independence of the square roots of distinct squarefree positive integers allows us to conclude that each coefficient of $h$ must be zero whereupon $f \in (x_1^2 - p_1, \cdots , x_N^2-p_N) \subset I$ follows, thereby yielding $I = \ker \phi$. The first isomorphism theorem now yields the above isomophism.

As a consequence of this, we see that the prime ideals of $R$ are in one-to-one correspondence with the prime ideals of $\mathbb Z[\{x_n\}_{n=1}^\infty]$ containing $I$. Now I also did observe that by the above argument itself any ideal $\mathfrak p \vartriangleleft \mathbb Z[\{x_n\}_{n=1}^\infty]$ containing $I$ can be generated by $\{x_n^2-p_n : n \in \mathbb N\}$ and squarefree monomials in the variables $\{x_n\}_{n=1}^\infty$. I tried to investigate what other linear combinations of squarefree monomials could a prime ideal $\mathfrak p$ contain, but such an investigation has gotten nowhere as of yet.

More precisely what I want to study are localizations of the ring $R$ at its prime ideals. I know that ideals of the localization at $\mathfrak p$ are in one to one correspondence with those of $A$ saturted by $A \setminus \mathfrak p$. So it again seems to come down to studying (as a first step) the prime ideals of the polynomial ring $\mathbb Z [\{x_n\}_{n=1}^\infty]$ containing $I$. Has this ring $R$ been studied before? I would really appreciate any references, proofs, hints, suggestions or ideas.

**Edit:** I vaguely recall reading somewhere a long time ago that all localizations of $R$ at prime ideals should be principal, so I would at least like to be able to prove that statement. Unfortunately, I can't recollect the reference, but if I remember correctly, this was stated there as a fact without any justification whatsoever.