Proposition 2

Let $A$ be a commutative ring, $\mathfrak{g}$ an ideal of $A$ and $B$ a commutative Noetherian $A$-algebra such that $B$ is contained in the Jacobson radical of $B$. Then every finitely generated $B$-module $M$ is an ideally Hausdorff $A$-module with respect to $\mathfrak{g}$.

We shall see more generally that for every finitely generated $A$-module $N$, $N \otimes_A M$ is Hausdorff with the $3$-adic topology. For $N_{(B)} = N \otimes_A B$ is a finitely generated $B$-module and the $B$-module $N \otimes_A M$ is canonically identified with $N_{,,} \otimes_B M$ by virtue of the associativity of the tensor product. Let $\mathfrak{L}$ be the Jacobson radical of $B$; as $\mathfrak{g} B$ is contained in $\mathfrak{L}$, the $\mathfrak{g}$-adic topology on $N \otimes_A M$ is therefore identified with a finer topology than the $\mathfrak{L}$-adic topology on $N_{(B)} \otimes_B M$; but this latter topology is Hausdorff since $N_{,,} \otimes_B M$ is a finitely generated $B$-module (no. 1, Example 1), whence the conclusion.