Transitivity of Finite Generation

Theorem
Let $\struct { C, +_C, \circ_C}$ be a ring with unity, where $1_C$ denotes the unity of $C$.

Let $\struct { B, +_B, \circ_B}$ be a subring of $C$, such that $1_C \in B$.

Let $\struct { A, +_A, \circ_A}$ be a subring of $B$, such that $1_C \in A$.

Let $B' = \struct { B, +_B, \circ_B}_A$ be a finitely generated $A$-module, where the scalar multiplication $\circ_B$ is defined as the ring product $\circ_B$ in $B$.

Let $C' = \struct { C, +_C, \circ_C}_B$ be a finitely generated $B$-module, where the scalar multiplication $\circ_C$ is defined as the ring product $\circ_C$ in $C$.

Then $C' ' = \struct { C, +_C, \circ_C}_A$ is a finitely generated $A$-module.

Proof
As $1_C \in B$, and $1_C \in A$, it follows that $A$ and $B$ are rings with unity.

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:


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

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


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

Then:

which shows that $x$ is a linear combination of $\set {b_j \circ_C c_k : j = 1, \ldots, n, k = 1, \ldots, m}$, with scalars $\alpha_{k j} \in A$ for all $k \in \set {1, \ldots, m}, j \in \set{1, \ldots, n}$.

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