Integral closure - Knowlpedia

Let $R$ be a commutative ring, and let $A$ be a commutative $R$ -algebra.

An element $a\in A$ is integral over $R$ if it satisfies a monic polynomial equation

a^n + r_{n-1}a^{n-1}+\cdots + r_1 a + r_0 = 0 \quad \text{with } r_i\in R.

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

\overline{R}^{\,A} \;=\; \{\, a\in A : a \text{ is integral over } R \,\}.

It is a subring of $A$ containing the image of $R$ .

When $R$ is a domain with fraction field $K$ , the integral closure of $R$ in $K$ is often called the normalization of $R$ . The domain $R$ is integrally closed precisely when its integral closure in its fraction field equals $R$ .

Properties

If $R\subseteq A$ is an integral extension, then every element of $A$ is integral over $R$ , hence $\overline{R}^{\,A}=A$ .
If $B\subseteq A$ is any subring containing $R$ and consisting of elements integral over $R$ , then $B\subseteq \overline{R}^{\,A}$ (maximality of the integral closure).

Examples

Integers inside rationals. Take $R=\mathbb{Z}$ and $A=\mathbb{Q}$ . If $a/b\in \mathbb{Q}$ (in lowest terms) is integral over $\mathbb{Z}$ , then it satisfies a monic polynomial with integer coefficients, forcing $b=\pm 1$ . Hence $\overline{\mathbb{Z}}^{\,\mathbb{Q}}=\mathbb{Z}$ .

A non-normal affine subring. Let $k$ be a field and consider $R=k[x^2,x^3]\subseteq A=k(x)$ (the rational function field in $x$ ). The element $x\in k(x)$ satisfies the monic equation $T^2-x^2=0$ with $x^2\in R$ , so $x$ is integral over $R$ . Thus the integral closure contains $k[x]$ . In fact one checks $\overline{R}^{\,k(x)}=k[x]$ .

Localization. If $R$ is a domain, $S\subseteq R$ is a multiplicative set of nonzero elements, and $\overline R$ is the integral closure of $R$ in its fraction field, then the integral closure of $S^{-1}R$ in that field is $S^{-1}\overline R$ . Thus integral closure commutes with localization.