Transitivity of Finite Generation

Theorem
Let $A$ be a ring with unity.

Let $B$ be a finitely generated $A$-module such that $B$ is also a ring with unity.

Let $C$ be a finitely generated $B$-module such that $C$ is also a ring with unity.

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

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

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

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


 * $\ds x = \sum_{k \mathop = 1}^m \beta_k c_k$

For any $k \in \set {1, \ldots, m}$, there exist $\alpha_{k j} \in A$, $j = 1, \ldots, n$ such that:


 * $\ds \beta_k = \sum_{j \mathop = 1}^n \alpha_{k j} b_j$

So


 * $\ds x = \sum_{k \mathop = 1}^m \sum_{j \mathop = 1}^n \alpha_{k j} b_j c_k$

Therefore, $\set {b_j c_k : j = 1, \ldots, n, k = 1, \ldots, m}$ generates $C$ over $A$.