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.