I have recently come upon an exercise in general ring theory whose main claim seems somewhat elusive. For reference, the exercise is $16. f)$, featured at the very end of section $\S 7$, chapter $8$ of Bourbaki's $Algebra$.
To make a precise statement of the claim in the exercise, consider an arbitrary ring $A$ and a Noetherian left $A$-module $M$. For arbitrary subgroups $G \leqslant A$ and $T \leqslant M$ we use the notation $G.T=\left\langle GT \right\rangle$ for the subgroup product between the two, i.e. the subgroup of $(M, +)$ generated by the subset product $GT=\left\{\lambda t\right\}_{\substack{\lambda \in G \\ t \in T}}$.
Write $\mathscr{Prim}(A)$ for the set of all primitive ideals of $A$ and $\mathscr{M}$ for the set of all finite products of primitive ideals of $A$ (which more technically can be expressed as the submonoid of $\left(\mathscr{Id}_{\mathrm{b}}(A), \bullet\right)$ - the monoid of all bilateral ideals of $A$ under ideal multiplication - generated by $\mathscr{Prim}(A)$). Under these conditions the claim is that the following intersection is the null submodule: $$\bigcap_{R \in \mathscr{M}}R.M=\left\{0_M\right\}.$$
The previous subexercise $16. e)$ - which is straightforward to solve - requires proving that if $N$ is a finitely generated left $A$-module such that $P.N=N$ for any $P \in \mathscr{Prim}(A)$ then $N$ is null, and the obvious indication (explicitly given by the authors) is to use this claim at part $e)$ in the context where the Noetherianity of $M$ ensures that all of its submodules are finitely generated. The immediate approach would therefore be that of considering the submodule $N=\displaystyle\bigcap_{R \in \mathscr{M}}R.M$ and of attempting to prove $P.N=N$ for any primitive ideal $P$. However, the main difficulty resides in the fact that the operations of multiplying by an ideal and taking intersections do not in general commute...
Am I missing something and is this where special properties of primitive ideals come into place so as for the above mentioned, generally lacking commutativity between the two operations to hold true in this particular instance? Or is it perhaps not directly to the intersection submodule $N$ that the claim of part $e)$ should be applied? Or is it simply the case that this result does not hold in general (and would only hold with additional conditions not included in the presentation of the exercise which I am referencing)?
If anyone has any knowledge of these matters I would be very grateful for some enlightenment.