Proposition 8

Let $A$ be a ring, $m$ a maximal ideal of $A$ and $M$ an $A$-module. If there exists an ideal $a$ of $A$ such that $m$ is the only maximal ideal of $A$ containing $a$ and $aM = 0$, then the canonical homomorphism $M \to M$, is bijective.

$A/a$ is then a local ring with maximal ideal $m/a$; $M$ can be considered as an $(A/a)$-module; for all $s \in A - m$ the canonical image of $s$ in $A/a$ is invertible, hence the homothety $x \mapsto sx$ of $M$ is bijective from the definition of $M$, as the solution of a universal problem ($§ 2$, no. 2); whence the proposition.

In particular, if there exists $k \geqslant 0$ such that $m^k M = 0$, the homomorphism $M \to M_m$ is bijective ($§ 1$, no. 1, Corollary to Proposition 1).