I'm trying to prove the following:
Let
$$
R = \left\{\frac{f(x)}{g(x)}\mid f(x),g(x)\in\mathbb{Q}[x], g(x)\not=0\right\}\subseteq\mathbb{Q}(x).
$$
Show that $R$ is a principal ideal domain.
My approach was to first prove that for any integral domain $S$ we have
$$
\begin{equation}
\text{Frac}(S[x])\cong\text{Frac}(S)(x)
\end{equation}
$$
and so $R\cong\mathbb{Q}(x)$, hence it's a field and so a PID. But I couldn't come up with a proof of this statement. Is this the right approach or is there a better way?
Best Answer
$\Bbb Q$ is a field $\implies \Bbb Q[X]$ is a PID. Then as $R=S^{-1}(\Bbb Q[X])$ where $S=\Bbb Q[X] \setminus \{0\}$ and localization of a PID is a PID, hence, $R$ is also a PID.