let $r^n+a_1r^{n-1}+\cdots+a_n=0$ be an equation of integral dependence of $r$ over an ideal $I$. does exist a finitely generated ideal $J$ subset of $I$ such that $a_i\in J^i$ for all $i=1,\ldots,n$?

Let $r^n+a_1r^{n-1}+\cdots+a_n=0$ be an equation of integral dependence of $r$ over an ideal $I$. Does exist a finitely generated ideal $J$, such that $J\subset I$ and $a_i\in J^i$ for all $i=1,\ldots,n$?

let $r^n+a_1r^{n-1}+...+a_n=0$ be an equation of integral dependence of $r$ over an ideal $I$. does exist a finitely generated ideal $J$ subset of $I$ such that $a_i\in J^i$ for all $i=1,...n$?

let $r^n+a_1r^{n-1}+\cdots+a_n=0$ be an equation of integral dependence of $r$ over an ideal $I$. does exist a finitely generated ideal $J$ subset of $I$ such that $a_i\in J^i$ for all $i=1,\ldots,n$?