Proposition 7.8 Atiyah Macdonald says that:
Let $A\subseteq B\subseteq C$ be rings. Suppose that $A$ is Noeterian, that $C$ is finitely generated as an $A$-algebra and that $C$ is either (i) finitely generated as a B-module or (ii) integral over $B$. Then $B$ is finitely generated as an $A$-module.
I try to solve it and here is my solution: It is easy to see $C$ is Noetherian as an $A$-module. On the other hand, $B$ is also an $A$-module (because $B$ is subring of $C$). So $B$ is Noetherian as an $A$-module and we have done.
I know it is wrong but I can not realise where it is. Can you help me point it out?