Integrally closed domain

Let $R$ be an integral domain with fraction field $K=\mathrm{Frac}(R)$.

Definition

$R$ is integrally closed if every element of $K$ that is integral over $R$ already lies in $R$. Equivalently,

where $\overline{R}^{\,K}$ is the integral closure of $R$ in $K$.

Remarks

• Many “nice” domains are integrally closed; for instance, every UFD is integrally closed (in particular, every PID is integrally closed).
• Integrally closed is one of the defining conditions of a Dedekind domain .

Examples

1. $\mathbb{Z}$.
   $\mathbb{Z}$ is integrally closed in $\mathbb{Q}$.

2. Polynomial rings.
   If $k$ is a field, then $k[x_1,\dots,x_n]$ is a UFD, hence integrally closed in its fraction field $k(x_1,\dots,x_n)$.

3. A non-example: $k[t^2,t^3]$.
   Let $R=k[t^2,t^3]\subset k(t)$. Then $t$ is integral over $R$ (since $t^2\in R$ and $t$ satisfies $x^2-t^2=0$) but $t\notin R$.
   Thus $R$ is not integrally closed; its integral closure in $k(t)$ is $k[t]$.