All rings $A$ between $\mathbb{Z}$ and $\mathbb{Q}$

Question

It's well know that if $S \subset \mathbb{Z}$ is a multiplicative such that $0 \notin S$, then $S^{-1}\mathbb{Z}$ is a ring between $\mathbb{Z}$ and $\mathbb{Q}$. I want to show that those rings are all. My idea is take $$S=\{b: \frac{a}{b} \in A; (a, b)=1 \}$$ for $A$ between $\mathbb{Z}$ and $\mathbb{Q}$, but I have had problems for show it's multiplicative.

Answer 1

If $\gcd(a,b)=1$ then $ua+vb=1$ for suitable $u,v\in \Bbb Z$, hence also $u\cdot \frac ab+v\cdot 1=\frac 1b\in A$. Thus $S=\{\,b\mid \frac 1b\in A\,\}$, and that is clearly multiplicative.