Proposition 5

Let $A$ be a commutative Noetherian ring, $m$ an ideal of $A$ and $E$ a finitely generated $A$-module. The closure $\bigcap_{n=1}^m m^nE$ *of $\{0\}$ in $E$ with respect to the $m$-adic topology is the set of $x \in E$ for which there exists an element $m \in m$ such that $(1 - m)x = 0$.

If $x = mx$ where $m \in m$, $x = m^n x \in m^nE$ for every integer $n \geq 0$ and hence $x \in F = \bigcap_{n=0}^m m^nE$. Conversely, if $x \in F$, $Ax$ is contained in the intersection of the neighbourhoods of 0 in $E$; it then follows from Theorem 2 that the $m$-adic topology on $Ax$, which is induced by that on $E$, is the coarsest topology; as $mx$ is by definition a neighbourhood of 0 with this topology, $mx = Ax$ and hence there exists $m \in m$ such that $x = mx$.

We may also say that $mF = F$ since

as $A$ is Noetherian, $F$ is a finitely generated $A$-module; it is then sufficient to apply Chapter II, § 2, no. 2, Corollary 3 to Proposition 4.