Let $R$ be a Noetherian 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 free finitely generated $A$-module. Is it true that $A$ is isomorphic to $R[x_1, \dots, x_n]$ as an ungraded $R$-algebra? Is this true for $n=2$ at least?
Examples I have in mind are $R[x_1^{i_1},\dots, x_n^{i_n}]\subset R[x_1,\dots, x_n]$ for $i_1, \dots, i_n\geq 1$ for which this is true.
One can note that $A$ has the same Krull dimension as $R[x_1,\dots, x_n]$.
EDIT: Under the assumptions $A$ is finitely presented. The answer should give an explicit presentation for $A$.
This is true when $n=2$ and the ground ring is an algebraically closed field of characteristic $0$ by https://projecteuclid.org/download/pdf_1/euclid.ojm/1200773129