How are these isomorphic $ K \otimes_{k_0} k \cong S^{-1} \left( R \otimes_{k_0} k \Big{/} P_0 \cdot (R \otimes_{k_0} k) \right)$

Question

Let $k_0$ be a field $k$ an algebraic closure of $k_0$. Let $R$ be a $k_0$ algebra and $P_0$ a prime ideal of $R$, $K$ is the fraction field of $R/P_0$. I am trying to understand how $$ K \otimes_{k_0} k \cong S^{-1} \left( R \otimes_{k_0} k \Big{/} P_0 \cdot (R \otimes_{k_0} k) \right), $$ as $k_0$-algebras, where $$ S = \{ z \otimes 1 + P_0 \cdot (R \otimes_{k_0} k): z \in R \backslash P_0 \}. $$

Any comments are appreciated!