Proposition 6

Let $(a_i)_{i \in I}$ be a non-empty finite family of ideals of a ring $A$ which are relatively prime in pairs; let $a$ be the intersection of the $a_i$. For every $A$-module $M$, the canonical mapping $M \to \prod_{i \in I} (M/a_i M)$ is surjective and its kernel is $aM$.

Clearly the canonical mapping of $M$ to $\prod_{i \in I} (M/a_i M)$ is zero on $aM$; then, by taking quotients, it defines a homomorphism $A : M/aM \to \prod_{i \in I} (M/a_i M)$. On the other hand, by Proposition 5, the canonical homomorphism

is bijective. Then so is $1_M \otimes \psi : M \otimes (A/a) \to M \otimes \prod_{i \in I} (A/a_i)$. Now $M \otimes (A/a)$ is identified with $M/aM$ and $M \otimes \prod_{i \in I} (A/a_i)$ with $\prod_{i \in I} M \otimes (A/a_i)$, which is itself identified with $\prod_{i \in I} (M/a_i M)$. It is immediately verified that the above identifications transforms $1_M \otimes \psi$ into $\lambda$, whence the proposition.