Remark 1

Conversely, let $M'_i$ be an $(A/b_i)$-module for each i, and consider the A-module M, the direct sum of the A-modules $M'_i$; then the submodules $M_i$ constructed above coincide with the $M'_i$ (it is enough to note that if $i \neq j$ then no nonzero element of $M'_j$ is annihilated by $b_i$ because $b_i + b_j = A$). Thus, roughly speaking, it amounts to the same thing to consider an A-module M or a family $(M_i)$ of modules over the rings $A/b_i = A_i$.