Finite-Type Algebra Property Under Change of Base

Question

I was wondering how to prove the following assertion. Any help would be appreciated!

Suppose we have to ring homomorphisms: $f: A \rightarrow B$ and $g: B \rightarrow C$.

Then $C$ being finite type over $A$ implies that $C$ is finite type over $B$.

Answer 1

$C$ being finite type over $A$ means that any element of $c\in C$ is of the form $$ c=\sum_{i=1}^ka_ix_i^{\alpha_i}$$ for $x_1,\ldots, x_k$ a fixed set of generators in $C$ and $a_i\in A$. Note that this above expression really means $$ c=\sum_{i=1}^k (g\circ f)(a_i)x_i^{\alpha_i}$$ by the definition of the algebra structure. In particular, if we set $f(a_i)=b_i$, we have $$ c=\sum_{i=1}^k g(b_i)x_i^{\alpha_i}$$ for $b_i\in B$. This is the same thing as $C$ being finite type over $B$ with generators $x_1,\ldots,x_k$: e.g. $$ c=\sum_{i=1}^k b_ix_i^{\alpha_i}.$$