Let $A \subseteq B \subseteq C$ be rings. I know that if $B$ is a finitely generated $A$-module and $C$ is a finitely generated $B$-module, then $C$ is a finitely generated $A$-module. (Proof is in Atiyah or here)
But I have some silly questions, which are some variances of the above. Let $A \subseteq B \subseteq C$ be rings. What can we say about $C$ ($C$ is a finitely generated $A$-module / $C$ is a finitely generated $A$-algebra / others) in the following cases?
- $B$ finitely generated $A$-module and $C$ finitely generated $B$-algebra
- $B$ finitely generated $A$-algebra and $C$ finitely generated $B$-module
- $B$ finitely generated $A$-algebra and $C$ finitely generated $B$-algebra
And in the same situation, is the follwing true?
a. $C$ is a finitely generated $A$-module implies $B$ is a finitely generated $A$-module and $C$ is a finitely generated $B$-module?
b. $C$ is a finitely generated $A$-algebra implies $B$ is a finitely generated $A$-algebra and $C$ is a finitely generated $B$-algebra?
Edit: I noticed that $A\subseteq B$ rings and $B=\sum_{i=1}^n A b_i$ is a finitely generated $A$-module implies $B$ is a finitely generated $A$-algebra, since $B=\sum_{i=1}^n A b_i=A[b_1,\cdots,b_n]$ by checking left, right inclusions. And I also solved (3), since if we let $B=A[b_1,\cdots,b_n], C=B[c_1,\cdots,c_m]$, then $C=(A[b_1,\cdots,b_n])[c_1,\cdots,c_m]=A[b_1,\cdots,b_n,c_1,\cdots,c_m]$ by checking both inclusions.
From this two facts, in questions (1), (2) we can deduce that $C$ is at least a finitely generated $A$-algebra. By using Amitesh's answer of (a), in (1), (3) $C$ need not be a finitely generated $A$-module otherwise $C$ should be a finitely generated $B$-module.
Then the left question is that:
In (2), what is the example of $C$ not being a finitely generated $A$-module?
In (a), decide whether or not $B$ needs to be a finitely generated $A$-module(or algebra).