# Definition:Integrally Closed

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## Definition

### Ring Extension

Let $\phi : A \hookrightarrow B$ be a ring extension.

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

Then $A$ is **integrally closed** in $B$ if and only if $C = \phi(A)$.

### Integral Domain

Let $R$ be an integral domain.

Then $R$ is **integrally closed** if and only if it is integrally closed in its field of fractions.