Let $R= \Bbb K [x]$ and $S=\{x^n: n \in \Bbb Z, n \geq 0 \} $. Let $D$ be the localization of $R$ in $S$, that is $D = S^{-1}R = \{ \frac{r}{s}: r \in R, s\in S \}$. By using the Universal Property of Localization prove that $S^{-1}R = \Bbb K [x,x^{-1}]$.
I've managed to prove the equality by using a standart isomorphism between the two rings but I can't quite seem to grasp how to apply the Universal Property of Localization. As far as I can understand, by using the UPL we come to a conclusion of the form $S^{-1}R \cong \Bbb K [x,x^{-1}]$. From that point it is trivial to show the equality. But how do we end up there?
Any insights will be duly appreciated. Thank you!