Integral dependence (also known as algebraic dependence) is a condition on ring extensions $R\subseteq S$: we say $s\in S$ is integral over $R$ if there is some $f(x) \in R[x]$ such that $f(s)=0$.
Integral dependence (also known as algebraic dependence) is a condition on ring extensions $$R\subseteq S$$: we say $$s\in S$$ is integral over $$R$$ if there is some $$f(x) \in R[x]$$ such that $$f(s)=0$$. The case where $$R,S$$ are fields forms the starting point of much of Galois theory.
An equivalent condition: $$s\in S$$ is integral iff there is a subring $$T\subseteq S$$ containing $$s$$ that is finitely-generated as an $$R$$-module. The idea is to use the (finite) powers of $$s$$ in $$f(s)$$ as generators.
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