Integral extension

Definition
Let $$R \le S$$ be commutative unital rings. We say that $$S$$ is an integral extension of $$R$$ if, for any element $$a \in S$$, there exists a monic polynomial $$p(x) \in R[x]$$ such that $$p(a) = 0$$.

Note that any integral extension is algebraic.

Metaproperties
Any integral extension of an integral extension is an integral extension.