Remark 3

The A-module M is cyclic if and only if each $M_i$ is cyclic: if $M = Am$, then $M_i = A_i e_i m$; conversely if $M_i = A_i m_i$ and $m = \sum_{i \in I} m_i$, then $M = Am$; indeed, if $n \in M$ projects onto $a_i m_i$ for each i, and if $a \in A$ is congruent to $a_i$ mod $b_i$ for each i, then am and n have the same image in each $M_i$, so coincide.