Remark

The hypothesis that $A$ is Noetherian is essential in this corollary. For example, let $A$ be the ring of infinitely differentiable mappings from $\mathbf{R}$ to itself and let $m$ be the (maximal) ideal of $A$ consisting of the functions $f$ such that $f(0) = 0$. It is immediate that $\bigcap_{n=0}^\infty m^n$ is the set of functions $f$ such that $f^{(n)}(0) = 0$ for all $n \geq 0$ and there exist such functions with $f(x) \neq 0$, for all $x \neq 0$, for example the function $f$ defined by $f(x) = e^{-1/x^2}$ for $x \neq 0$ and $f(0) = 0$.