An $I$-adic noetherian ring is a noetherian ring which is separated and complete in the I-adic topology for some ideal $I ⊂ A$.
Let $p_1$ and $p_2$ are two prime ideals of $A$ which contain $I$, and let $n,m$ be two positive integers such that $n<m$. Fixed an element $x_1\in p_1$, if there exists some element $x_2\in p_2$ such that $x_1-x_2\in I^n$, does there exist an element $x_3$ in $p_2$ such that $x_1-x_3\in I^m$ ？
Thanks in advance !