Can you tell me if my reasoning is correct?
I want to prove if $S \subset R$ are rings and $R$ is integral over $S$ and $I$ is an ideal of $R$ then $R/I$ is integral over $S/ (S\cap I)$.
Let $R$ be integral over $S$. $(S \cap I) \subset I$ is an ideal of $S$ and hence of $R$.
Since $R$ is integral over $S$ we have that $R/(S \cap I)$ is integral over $S/(S \cap I)$ and since $(S \cap I) \subset I$ we have $R/I \subset R/(S\cap I)$ and hence $R/I$ is integral over $S/(S\cap I)$.