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 $\mathbb{C}[x_1,\ldots, x_n]$?
It is not possible.
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 $x_i\in A$.
