Let $(R, \mathfrak m)$ be a Noetherian local ring of dimension $d>0$. Let $I$ be an $\mathfrak m$-primary ideal of $R$ i.e. $\sqrt I =\mathfrak m$ .
How to show that there exists $x_1,...,x_d \in I$ such that for some $n \ge 1$, we have $I^{n+1}=I^n (x_1,...,x_d)$ ?
I am not even able to do the case $d=1$.
I know that for some $x_1,...,x_d\in R$ , we have $\sqrt {(x_1,...,x_d)}=\mathfrak m$, but that's not enough.
Please help.