Let $A\subseteq B\subseteq C$ be commutative unital rings. Recall that the extension $A \subseteq B$ is finite / of finite type / integral, when $B$ is a finitely generated $R$-module / when $B$ is a finitely generated $A$-algebra / when $\forall b \in B$ $\exists$ monic polynomial $f \in A[x]$ with $f(b)=0$. Notation $_AB$ means "$A$-module $B$".
We know that $A\subseteq C$ is integral iff $A\subseteq B$ and $B\subseteq C$ are integral (Grillet, Abstract Algebra, 7.3.3).
Do we also have the following:
- $A\subseteq C$ is finite $\Leftrightarrow$ $A\subseteq B$ and $B\subseteq C$ are finite.
- $A\subseteq C$ is of finite type $\Leftrightarrow$ $A\subseteq B$ and $B\subseteq C$ are of finite type.
I'm having problems with ($A\subseteq C$ finite $\Rightarrow$ $A\subseteq B$ finite) and with ($A\subseteq C$ of finite type $\Rightarrow$ $A\subseteq B$ of finite type). If $_AB$ is a direct summand of $_AC$ (for example when $A$ is a field), i.e. $_AC= _A B\oplus _A B'$ for some submodule $B'$ of $C$, then $C=Ac_1+\cdots+Ac_n$ implies $c_i=b_i+b'_i$ for some $b_i\in B$ and $b'_i \in B$, hence $B = Ab_1 + \cdots + Ab_n$. But what if $_AB$ is not a direct summand of $_AC$? And what about the 'finite type' case?