Exercise 1, § 5

Let $A$ be a commutative ring and $3$ an ideal of $A$. An $A$-module $M$ is called absolutely Hausdorff with respect to $3$ if, for every finitely generated $A$-module $N$, the $A$-module $N \otimes_A M$ is Hausdorff with the $\mathfrak{g}$-adic topology; an absolutely Hausdorff $A$-module is ideally Hausdorff.

(a) For $M$ to be absolutely Hausdorff with respect to $\mathfrak{g}$, it is necessary and sufficient that, for every finitely generated $A$-module $N$ and every submodule $N'$ of $N$, $\operatorname{Im}(N' \otimes_A M)$ be closed in $N \otimes_A M$ with the $3$-adic topology.

(b) Let $B$ be a commutative $A$-algebra and $\mathfrak{L}$ an ideal of $B$ containing $\mathfrak{g}B$. If $M$ is an absolutely Hausdorff $B$-module with respect to $\mathfrak{L}$, $M$ is an absolutely Hausdorff $A$-module with respect to $3$.