Integrally Closed is Local Property

Theorem

Let $A$ be an integral domain.

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

Then the following are equivalent:

1. $A$ is integrally closed

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

3.$A_{\mathfrak m}$ is integrally closed for all maximal ideals $\mathfrak m$.

Proof

1. $\Rightarrow$ 2.

Let $Q(R)$ denote the quotient field of a domain $R$.

Since $Q(A_{\mathfrak p}) = Q(A)$, by Localisation Preserves Integral Closure, we have that $A_{\mathfrak p}$ is integrally closed for all prime ideals $\mathfrak p$.

2. $\Rightarrow$ 3.

This is true because a Maximal Ideal is Prime.

3. $\Rightarrow$ 1.