Definition:Integrally Closed

Ring Extension
Let $R \subseteq A$ be an extension of a commutative ring with unity.

Let $C$ be the integral closure of $R$ in $A$.

If $C = R$ then $R$ is said to be integrally closed in $A$.

Integral Domain
If $R$ is an integral domain, then $R$ is integrally closed if it is integrally closed in its quotient field.