Transitivity of Finite Generation

Theorem

Let $A \subseteq B \subseteq C$ be rings.

Suppose $B$ is a finitely generated $A$-module, and $C$ is a finitely generated $B$-module.

Then $C$ is a finitely generated $A$-module.

Proof

Let $b_1,\ldots, b_n$ generate $B$ over $A$.

Let $c_1,\ldots, c_m$ generate $C$ over $B$.

Then for any $x \in C$ there are $\beta_k \in B$, $k=1,\ldots, m$ such that:

$\displaystyle x = \sum_{k = 1}^m \beta_k c_k$

For any $i \in \{1,\ldots,m\}$ there exists $\alpha_{ij} \in A$, $j = 1,\ldots, n$ such that

$\displaystyle \beta_i = \sum_{j = 1}^n \alpha_{ij}b_j$

So

$\displaystyle x = \sum_{k = 1}^m \sum_{j = 1}^n \alpha_{kj} b_j c_k$

Therefore, $\{b_jc_k : j = 1,\ldots,n;\ k = 1,\ldots, m\}$ generates $C$ over $A$.

$\blacksquare$