Let $R$ be a commutative unital ring. Let $n\geq 1$ be an integer. Suppose that $R[x_1, \dots, x_n]$ has a $R$-subalgebra $A$ such that $R[x_1, \dots, x_n]$ is a finitely generated $A$-module. Is it true that $R[x_1, \dots, x_n]$ is a free $A$-module? Is this true for $n=2$ at least?
I think so because there are no non-trivial relations satisfied by $x_1,\dots, x_n$ but I am not sure what happens when there is torsion.