Proposition 11

Let $n_1, \ldots, n_d$ be integers $\geqslant 1$ which are relatively prime in pairs and $n = n_1 \ldots n_r$. The canonical homomorphism of $\mathbf{Z}$ into the product ring $\prod_{i=1}^{r} \mathbf{Z}/n_i\mathbf{Z}$ is surjective of kernel $n\mathbf{Z}$ and defines a ring isomorphism of $\mathbf{Z}/n\mathbf{Z}$ onto $\prod_{i=1}^{r} \mathbf{Z}/n_i\mathbf{Z}$.

Let $a_i = n_i\mathbf{Z}$ for $i = 1, \ldots, r$. By hypothesis, $a_i + a_j = \mathbf{Z}$ for $i \neq j$. The proposition then follows from Proposition 9.

The above results, as also those concerning decomposition into prime factors, will be generalized in Chapter VII, § 1, which is devoted to the study of principal ideal domains and in Commutative Algebra, Chapter VII, § 3, which is devoted to the study of factorial domains.