I have searched the site, but there is a lot of confusion regarding this topic, so I try to be as clear as possible.
Let $R$ be an integral domain and denote with $K(R)$ the field of fractions of $R$.
Now, let $S\subseteq R$, $0\notin S$, any multiplicatively closed subset of $R$.
Let $S^{-1}R$ be the localization of $R$ at $S$. Suppose that $S^{-1}R$ is a integral domain.
Is the following true or false?
$$K(R)\cong K(S^{-1}R)$$
Can this isomorphism be derived from the universal property of localization? If so, how can I proceed strictly?