Proposition 1

The $A$-module $M$ is the direct sum of its submodules $M_i$.

Let $p_i : M \to M$ denote the homothety $m \mapsto e_i m$; the map $p_i$ is $A$-linear since $e_i$ is central; since $e_i^2 = e_i$, $\sum_{i \in I} e_i = 1$ and $e_i e_j = 0$ for $i \neq j$, we have

and the $p_i$ form an orthogonal family of projectors whose sum is the identity (II, p. 209, Def. 7). By loc. cit., Prop. 12, the module M is the direct sum of the submodules $p_i(M) = e_iM$. In addition $e_iM$ is annihilated by $b_i = A(1 - e_i)$; if $i \neq j$ and $m \in M$ then $(1 - e_i)e_jm = e_jm$, whence no nonzero element of $e_jM$ is annihilated by $1 - e_i$, and so a fortiori by $b_i$. It follows that $e_iM = M_i$, and the proposition is proved.