Corollary

If M is a module of finite length over a Noetherian ring A, then

This will follow from Proposition 8 (iv) if we prove that

Now, it follows from Proposition 1 of no. 1 that for all $s \in A - p$ the homothety with ratio s on $M/\mathscr{Q}(p)$ is injective; the homothety with ratio s on every submodule R of $M/\mathscr{Q}(p)$ is therefore injective and, as R is Artinian, it is bijective (Algebra, Chapter VIII, § 2, no. 2, Lemma 3); we conclude that the sub-A-modules of $M/\mathscr{Q}(p)$ are the images under the bijection $f : M, \to M/\mathscr{Q}(p)$ of the sub-A,-modules of M, (Chapter 11, § 2, no. 3), whence our assertion.