In Hungerford’s Algebra, there is a problem which asks for conditions on $S$, a multiplicative subset of $\mathbb{Z}$, which ensure an isomorphism between $\mathbb{Q}$ and $S^{-1} \mathbb{Z}$.
Examples of multiplicative subsets which yield the result are $n \mathbb{Z}$ for $n > 1$ and an example of a multiplicative subset which does not yield the rationals is $\{ 2^k | k \in \mathbb{N} \}$. The multiplicative subsets just mentioned are relatively simple though and so I’m not exactly sure what condition(s) on $S$ would be vital in this context.
$S^{-1} \mathbb{Z}$ is the smallest commutative ring for which each element of $S$ is a unit and so there must be a copy of this ring within $\mathbb{Q}$ and if $S^{-1} \mathbb{Z}$ is a field then it must contain a copy of $\mathbb{Q}$. So, would it be enough to find a condition on $S$ so that $S^{-1} \mathbb{Z}$? More so, what is such a condition?