A subring $R$ of a commutative ring $S$ is said to be integrally closed in $S$ if whenever $\theta \in S$ and $\theta$ is integral over $R$ then $\theta \in R$

The integral closure of $R$ in $S$ is integrally closed in $S$

An integral domain $R$ is said to be integrally closed (or normal) if it is integrally closed in its fraction field.