Hello, maybe this is a naive question, but so far I did not found anything related to the subject.

I would like to consider a subset of integers, say E, such that the set $\{ \frac{x}{y}, x \in E, y \in E, y \neq 0 \}$ is $\mathbb{Q}$.

Do such sets have a particular name? Is anyone known for having studied them? And is it possible to define such a set for which any (positive or non-zero) rational is uniquely represented as a ratio of elements in $E$?

Thanks by advance for your comments!