I'm currently in an algebraic geometry class and many of our proofs and lemmas have to do with some element $w$ being integral over a ring. The proofs end up working out, but I have trouble understanding why this is an important property to have (being the root of a monic polynomial) and what intuition I should have behind it. For example, consider:
$R \subset F, \space L=Frac(R). Let \space w \in I$ be algebraic over $R$. Then $\exists \space p \in R,p≠0$ s.t. $pw$ is integral over R.