Integral extension

Definition (integral extension)

Let $R \subseteq S$ be a ring extension (commutative, with $1$). The extension is integral if every element $s\in S$ is integral over R , i.e. each $s$ satisfies some monic polynomial in $R[t]$.

Useful equivalent criteria

• If $S$ is finitely generated as an $R$-module , then $S$ is integral over $R$. (A “finite” extension is automatically integral.)
• If $S = R[s_1,\dots,s_n]$ and each $s_i$ is integral over $R$, then $S$ is a finite $R$-module (so in particular the extension is integral).

Integral extensions have strong prime-ideal behavior; key theorems include lying over and going up .

Examples

1. $\mathbb Z \subset \mathbb Z[i]$.
   Every element $a+bi\in \mathbb Z[i]$ is integral over $\mathbb Z$ because it satisfies a monic polynomial over $\mathbb Z$ (e.g. $t^2-2at+(a^2+b^2)=0$). Hence $\mathbb Z[i]$ is integral over $\mathbb Z$.

2. A hypersurface example.
   Let $k$ be a field and consider

   $$R = k[x] \subset S = k[x,y]/(y^2-x).$$

   The class of $y$ in $S$ satisfies $t^2-x=0$, a monic polynomial in $R[t]$. Thus $S$ is integral over $R$.

3. Non-example: $\mathbb Z \subset \mathbb Q$.
   The extension is not integral, since $1/2\in \mathbb Q$ is not integral over $\mathbb Z$ (see integral element ).

Related knowls

• Elementwise notion: integral element
• Integral closure: integral closure , integrally closed domain
• Prime behavior: lying over theorem , going up theorem , going down theorem
• Spectrum viewpoint: prime spectrum