If $A<C$ is finite must $A<B,B<C$ be finite

Question

Let $A<B<C$ be a tower of commutative rings with identity. If $C$ is finitely generated as an $A$-module, must we have the following:

1) $B$ is finitely generated as an $A$-module.

2) $C$ is finitely generated as an $B$-module. (This is trivial already as noted by user26857.)

Do 1), 2) have to hold if at least we restrict ourselves to integral domains?

Thank you.