Exercise 4, § 5

Let $A$ be a commutative ring, $3$ an ideal of $A$ and $S = 1 + 3$. Show that, if $M$ is an absolutely Hausdorff $A$-module with respect to $\mathfrak{g}$, then $S^{-1}M = M$, in other words, for all $a \in S$, $x \mapsto ax$ is a bijection of $M$ onto itself. (Show first that the hypothesis that $M$ is Hausdorff with the $3$-adic topology implies that $x \mapsto ax$ is injective; then prove that the submodule $aM$ of $M$ is dense in $M$ with the $8$-adic topology and use Exercise 1 (a).)