Definition 10.35.9. Let $R \to S$ be a ring map. The ring $S' \subset S$ of elements integral over $R$, see Lemma 10.35.7, is called the integral closure of $R$ in $S$. If $R \subset S$ we say that $R$ is integrally closed in $S$ if $R = S'$.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).