Corollary

Let $A$ be a ring, $m_1, m_2, \ldots, m_n$ distinct maximal ideals of $A$, $M$ an $A$-module and $k_1, k_2, \ldots, k_n$ integers $\geqslant 0$. The canonical homomorphism from $M$ to $\prod^n M_m/m_{i_1}^{k_1} M_m$ is surjective and its kernel is $\left( \bigcap^n m_{i_1}^{k_1} \right) M$.

This follows easily from Proposition 9 and $§ 1$, no. 2, Proposition 6, the $m_{i_1}^{k_1}$ being relatively prime in pairs ($§ 1$, no. 2, Proposition 3).

In the rest of this no. $A$ will denote a ring and $\Omega(A)$ (or $\Omega$) the set of maximal ideals of $A$.