Proposition 14

Let $e$ be an idempotent in A.

a) If $(e_i)_{i\in I}$ is a partition of $e$, then the A-module $Ae$ is the direct sum of the family $(Ae_i)_{i\in I}$.

b) Let $(\mathfrak{a}_i)_{i\in I}$ be a finite family of left ideals of A with direct sum $Ae$. For $i\in I$, denote the component of $e$ in $\mathfrak{a}_i$ by $e_i$. Then $(e_i)_{i\in I}$ is a partition of $e$, and we have $\mathfrak{a}_i= Ae_i$ for every $i\in I$.

c) The A-module $Ae$ is indecomposable if and only if the idempotent $e$ is indecomposable.

Let $(e_i)_{i\in I}$ be a partition of $e$. For every $i\in I$, we have

and therefore $Ae_i\subset Ae$. For every $i\in I$, we define an A-linear projector $p_i$ in $Ae$ by setting $p_i(x) =xe_i$. We have $p_ip_j= 0$ if $i\not=j$, and for every $x$ in $Ae$,

Consequently (II, §1, No. 8, p. 20, Proposition $12$)$, Ae$ is the direct sum of the images of the $p_i$. Now, we have $ee_i=e_ie=e_i$, so the image of $p_i$ is $Ae_i$. This proves a).

Take the notation and assumptions of b). Let $i\in I$. Since $e_i$ belongs to $Ae$, we have $e_i=e_ie=\sum_je_ie_j$. Since $Ae$ is the direct sum of the $\mathfrak{a}_j$ and $e_ie_j$ belongs to $\mathfrak{a}_j$ for every $j$, we have $e_i=e_ie_i$ and $e_ie_j= 0$ for $i\not=j$. In other words, $(e_i)_{i\in I}$ is a partition of the idempotent $e$. By a), $Ae$ is the direct sum of the $Ae_i$. By assumption, we have $Ae_i\subset \mathfrak{a}_i$, and $Ae$ is the direct sum of the $\mathfrak{a}_i$. We therefore have $Ae_i=\mathfrak{a}_i$ for every $i\in I$. We have proved b).

Finally, c) follows immediately from a) and b).