Integrally Closed is Local Property

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Theorem

Let $A$ be an integral domain.

For a prime ideal $\mathfrak p$ of $A$, let $A_{\mathfrak p}$ denote the localization at $S = A \divides \mathfrak p$.

Then the following are equivalent:

$(1): \quad A$ is integrally closed
$(2): \quad A_{\mathfrak p}$ is integrally closed for all prime ideals $\mathfrak p$.
$(3): \quad A_{\mathfrak m}$ is integrally closed for all maximal ideals $\mathfrak m$.


Proof

$(1)$ implies $(2)$

Let $\map Q R$ denote the quotient field of an integral domain $R$.

We have by Localization Preserves Integral Closure that:

$\map Q {A_{\mathfrak p} = \map Q \A$

Hence $A_{\mathfrak p}$ is integrally closed for all prime ideals $\mathfrak p$.

$\Box$


$(2)$ implies $(3)$

This is true because a Maximal Ideal of Commutative and Unitary Ring is Prime Ideal.

$\Box$


$(3)$ implies $(1)$