Proposition 4

Let $(a_i)$, $i \leq n$ be a finite family of pairwise coprime elements of the principal ideal domain $A$. Then the canonical homomorphism (I, p. 110) from $A / \left( \prod_{r=1}^n a_r \right)$ into the product $\prod_{r=1}^n A/(a_r)$ is an isomorphism of $A$-algebras.

This follows from I, p. 110, Prop. 9 and from Prop. 2 a).

The conclusion of Prop. 4 is not valid if it is not assumed that the $a_i$ are pairwise coprime (cf. VII, p. 24, Prop. 9).