Let $R$ be a principal ideal domain, and $S \subseteq Q(R)$ a subring of the quotient field of $R$, so that $R \subseteq S$. I want to show that, for any $x, y \in R$:
$$\frac{x}{y} \in S \implies \frac{y}{\gcd(x, y)} \in S^\times$$
where $S^\times$ is the group of units of $S$, as well as that if $I \subseteq S$ is an ideal of $S$, then $R \cap I$ is an ideal of $R$. Also, why is $S$ a principal ideal domain?
Thanks in advance. I'm not so used to working with quotient fields. I've especially been struggling with the first statement so far.
Edit: sorry for the confusion, edited it to make it more clear. Well if $\frac{y}{\gcd(x, y)} \in S^\times$, then its reciprocal is too by definition of $S^\times$.
Let $d := \text{gcd}(x,y)$. Then there are $a,b\in R$ with $ax+by = d$, and hence $\frac dy = a\frac xy + b \in S$.
If $I \subseteq S$ is an ideal, it should be easy to check that $I\cap R$ is an ideal of $R$. Is there a particular axiom you're having trouble with? Certainly, once we know this, then $I \cap R = zR$ for some $z\in R$, and we claim that $I = zS$. Indeed, $z\in I$. Conversely, if $\frac xy \in I$, then $x = y\frac xy \in I$ as well. Therefore, we can write $x = za$ for some $a\in R$, and we conclude that $\frac xy = z\frac ay \in zS$.