Definition 6.6 (Integral closure). Let $R$ be a subring of $S$. We say $s\in S$ is integral over $R$ if there exists a monic polynomial $f(X)\in R[X]$ such that $f(s)=0$.

The integral closure $R^{int(S)}$ of $R$ inside $S$ is defined to be

$$R^{int(S)}=\{s\in S|s\text{ integral over }R\}.$$

We say $R$ is integrally closed in $S$ if $R^{int(S)}=R$.