Let $R = \mathbb{Z}[\sqrt{-5},1/2,1/5]$. Inverting the ramified primes $2,5$ simplifies the proof that every maximal ideal is inversible ie. the unique factorization in maximal ideals. In $O_K=\mathbb{Z}[\sqrt{-5}]$ for any ideal there is $c \in I, N(c) \le 6 N(I)$, the latter stays true for any ideal $I$ in $R$, thus $(c) = IJ, N(J) \le 6$, since $N(1+\sqrt{-5}) = 3$ the ideals of norm $\le 6$ are principal, whence $J = (b)$ is principal and $I=(c/b)$ and $R$ is a PID.
Is it correct ? Do you know some resources on those kind of rings ? Are there some troubles in using those rings of $\Bbb{Z}[1/q]$-integers instead of $O_K$ when working with number fields ? Do they help for something, say Diophantine equations ?