Integral element

Definition (integral over a ring)

Let $R \subseteq S$ be a ring extension (typically commutative, with $1$). An element $x\in S$ is integral over $R$ if there exists a monic polynomial

such that $f(x)=0$.

Equivalently, $x$ is integral over $R$ iff the subring $R[x]\subseteq S$ is a finitely generated $R$-module (in fact generated by $1,x,\dots,x^{n-1}$).

An extension $R\subseteq S$ is called integral if every element of $S$ is integral over $R$.

Examples

1. Gaussian integers.
   In $\mathbb Z \subset \mathbb Z[i]$, the element $i$ is integral over $\mathbb Z$ since it satisfies the monic polynomial

   $$t^2+1=0.$$

2. Adjoining a square root.
   In $\mathbb Z \subset \mathbb Z[\sqrt{2}]$, the element $\sqrt{2}$ is integral over $\mathbb Z$ since it satisfies

   $$t^2-2=0.$$

3. A non-example: $1/2$ over $\mathbb Z$.
   The element $1/2\in \mathbb Q$ is not integral over $\mathbb Z$.
   Indeed, if $(1/2)$ satisfied a monic polynomial with integer coefficients, multiplying by a suitable power of $2$ would force an integer divisibility contradiction (equivalently: the only rationals integral over $\mathbb Z$ are the integers).

Related knowls

• Integral extensions: integral extension
• Integral closure: integral closure , integrally closed domain
• Rings and modules: commutative ring , module