how can i prove that every subring of $\mathbb{Q}$ is PID?

Question

How can I prove that every subring of $\mathbb{Q}$ is PID?

Answer 1

Let $R$ be a subring of $\mathbb{Q}$. Then the function $f(a/b) := |a|$ is a Euclidean function on $R$. In particular, $R$ is a PID. (See also.)

A more general result has been linked, but deduction from Euclidean to PID is probably much easier in this case.