Corollary (Krull)

Let $A$ be a commutative Noetherian ring and $m$ an ideal of $A$. The ideal $\bigcap_{n=1}^m m^n$ is the set of elements $x \in A$ for which there exists $m \in m$ such that $(1 - m)x = 0$. In particular, for $\bigcap_{n=1}^\infty m^n = \{0\}$, it is necessary and sufficient that no element of $1 + m$ be a divisor of $0$ in $A$.

It is sufficient to apply Proposition 5 to $E = A$.