I've recently came across the notion of a Ring $B$ being integral over a subring $A$. Atiyah-MacDonald (at least the few pages right after the definition) refrain from explaining the intuitive interpretation of the definition.
Why do we define the "integral ring over" relationship?
Does this relationship captures some important properties of the "naive" ring of integers $\mathbb{Z}$? Maybe properties of the relationship with other rings $\mathbb{Z}$ is traditionally embedded in?
What are those properties?