Proposition 9

Let $A$ be a ring, $m$ a maximal ideal of $A$, $M$ an $A$-module and $k$ an integer $\geqslant 0$. The canonical homomorphism $M \to M_m/m^k M_m$ is surjective, has kernel $m^k M$ and defines an isomorphism of $M/m^k M$ onto $M_m/m^k M_m$.

Since the case $k = 0$ is trivial, suppose that $k \geqslant 1$. It follows from Proposition 8 that the canonical homomorphism $M/m^k M \to (M/m^k M)_m$ is bijective. On the other hand $(M/m^k M)_m$ is canonically identified with $M_m/(m^k M)_m$ (§ 2, no. 4, Theorem 1) and hence $(m^k M)_m = m^k M_m$ (§ 2, no. 7, Corollary to Proposition 18), whence there is an isomorphism of $M/m^k M$ onto $M_m/m^k M_m$ which maps the class of an element $x \in M$ to the class of $x/1$.