# Big polynomial subalgebra of polynomials

Consider some algebraically independent polynomials $$f_1,\ldots, f_n\in\mathbb{C}[x_1,\ldots, x_n]$$.

Is it possible that $$I\subseteq\mathbb{C}[f_1,\ldots, f_n]\subsetneq\mathbb{C}[x_1,\ldots, x_n]$$ for some not trivial ideal $$I$$ of $$\mathbb{C}[x_1,\ldots, x_n]$$?

First, note that in the situation, $$0\neq I\subset A=\mathbb{C}[f_1,\ldots, f_n]\subset B=\mathbb{C}[x_1,\ldots, x_n]$$ with $$I$$ an ideal of $$B$$, one must have $$A$$ and $$B$$ birational. To see this, let $$0\neq p\in I$$. Then, $$x_ip\in I$$ and thus, $$x_i=x_ip/p$$.
Next, assume that $$x_i\not\in A$$. Write $$x_i=F/G$$ with $$F,G\in A$$ and coprime in $$A$$, since $$A$$ is a polynomial ring and thus a UFD. Then, $$x_i^mp\in I$$ and so, $$F^mp/G^m\in A$$. This says, $$G^m$$ divides $$p$$ in $$A$$ for all $$m$$ and the only way this can happen is if $$G=1$$ and so $$x_i\in A$$.