Say we have $\mathbb{Q}[\sqrt{d}] = \{a+b\sqrt{d} | a,b\in\mathbb{Q}\}$ and $\mathbb{Z}[\sqrt{d}] = \{a+b\sqrt{d} | a,b\in\mathbb{Z}\}$ for square-free integers $d$. I want to show that $\mathbb{Q}[\sqrt{d}]=\mathbb{Z}[\sqrt{d}]_S$ (here $S$ is the set of multiplicative nonzero integers).
In my head it logically makes sense, but I cannot think of how to formalize this proof. Any help is appreciated!