Let $S$ be a subring of $R$ such that $R$ is integral over $S$. Let $P$ be a prime ideal of $S$ and $M=S-P$. Let $S_M$ be the quotient ring of $S$ and $R_M$ the quotient ring of $R$. Let $i: S \to S_M, j: R\to R_M$ be the natural maps and $P_M=i(P)S_M$. It is said that $$i^{-1}(P_M)=P. \quad (1)$$
Suppose that $P^*\subset R_M$ is a prime ideal of $R_M$ such that $P^* \cap S_M = P_M$. It is said that $$j^{-1}(P^*) \cap S = i^{-1}(P^*\cap S_M). \quad (2)$$ How to show the results (1) and (2)? These questions are from page 2 of the Red Book of Varieties and Schemes. Thank you very much.
