Can Q(R) embed to Q((R ⊗ S )/ P)

Question

Let $R, S$ be Noetherian $k$-algebra, where $k$ is a field, and $P \otimes S$ is Noetherian.

let $P$ be a prime ideal of $R \otimes S$ such that $P \cap (R \otimes 1) = (0) = P \cap (1 \otimes S)$, then it is obvious $R \hookrightarrow (R \otimes S)/P$

Can $Q(R) \hookrightarrow Q((R \otimes S) / P)$ ? where $Q(R)$ is classical quotient ring of $R$