Integral closure

Let $R \subseteq A$ be commutative rings.

Definition

An element $a \in A$ is integral over $R$ if there exists a monic polynomial

(See integral element .)

The integral closure of $R$ in $A$ is

Equivalently, $\overline{R}^{\,A}$ is the largest subring $B$ with $R \subseteq B \subseteq A$ such that $B$ is an integral extension of $R$.

When $R$ is an integral domain with fraction field $K$, the phrase “integral closure of $R$” often means the integral closure of $R$ in $K$. In that case, $R$ is integrally closed iff $\overline{R}^{\,K}=R$.

Examples

1. $\mathbb{Z}\subset \mathbb{Q}$.
   The integral closure of $\mathbb{Z}$ in $\mathbb{Q}$ is $\mathbb{Z}$: the only rationals integral over $\mathbb{Z}$ are the integers.

2. $\mathbb{Z}\subset \mathbb{Q}(\sqrt{5})$.
   The integral closure of $\mathbb{Z}$ in $\mathbb{Q}(\sqrt{5})$ is

   $$\mathbb{Z}\Big[\tfrac{1+\sqrt{5}}{2}\Big],$$

   since $\frac{1+\sqrt{5}}{2}$ satisfies the monic equation $x^2-x-1=0$.

3. A cusp subring.
   Let $k$ be a field and $R=k[t^2,t^3]\subset k[t]\subset k(t)$.
   Then $t$ is integral over $R$ because it satisfies $x^2-t^2=0$ with $t^2\in R$. One checks that the integral closure of $R$ in $k(t)$ is $k[t]$.