Change of variables in $k$-algebras Suppose $k$ is an algebraically closed field, and let $I$ be a proper ideal of $k[x_1, \dots, x_n]$.  Does there exist an ideal $J \subseteq (x_1, \dots, x_n)$ such that $k[x_1, \dots, x_n]/I \cong k[x_1, \dots, x_n]/J$ as $k$-algebras?

This is very clear from the corresponding geometric statement: Every non-empty algebraic set $\subseteq \mathbb{A}^n$ can be moved to some that contains the zero. Actually a translation from some point to the zero suffices.
