Definition:Integral Dependence

Definition

Let $A$ be a commutative ring with unity.

Let $R \subseteq A$ be a subring.

Then $a \in A$ is said to be integral over $R$ if it satisfies an equation of the form

$a^n + r_{n-1}a^{n-1} + \cdots + r_1 a + r_0 = 0$

for some $r_i \in R$, $n \in \N$.

The ring extension $R \subseteq A$ is said to be integral if for all $a \in A$, $a$ is integral over $R$.