Finitely generated over a ring $S$ and a map $R \to S$ implies finitely generated over $R$?

Question

Let $R, S$ be commutative rings. Suppose I have a finitely generated $S$-algebra $A$. Let $\phi: R \to S$ be a ring homomorphism. Then does this turn $A$ into a finitely generated $R$-algebra? Thank you!

Answer 1

Given $R,S$, this is true for all $A$ if and only if $S$ is finitely generated over $R$.

Indeed if it's true for all $A$, then it's true for $A=S$, so $S$ is finitely generated over $R$.

Conversely, if $x_1,...,x_n$ generate $S$ over $R$ and $a_1,...,a_k$ generate $A$ over $S$, then $x_1\cdot 1_A,...,x_n\cdot 1_A, a_1,...,a_k$ generate $A$ over $R$.