Exercise 12, § 8

Let $A$ be a ring and $(a_i)_{i \in I}$ a finite family of left ideals of $A$ such that the additive group of $A$ is the direct sum of the additive groups $a_i$. If $1 = \sum_{i \in I} e_i (e_i \in a_i)$, then $e_i^2 = e_i, e_i e_j = 0$ if $i \neq j$ and $a_i = Ae_i$ (write $x = x . 1$ for all $x \in A$). Conversely, if $(e_i)_{i \in I}$ is a finite family of idempotents of $A$ such that $e_i e_j = 0$ for $i \neq j$ and $1 = \sum_i e_i$, then $A$ is the direct sum of the left ideals $Ae_i$. For the ideals $Ae_i$ to be two-sided, it is necessary and sufficient that the $e_i$ be in the centre of $A$.