Exercise 13, § 8

Let $A$ be a ring and $e$ an idempotent of $A$. (a) Show that the additive group of $A$ is the direct sum of the left ideal $a = Ae$ and the left annihilator $b$ of $e$ (note that, for all $x \in A, x - xe \in b$). (b) Every right ideal $d$ of $A$ is the direct sum of $d \cap a$ and $d \cap b$. (c) If $Ae = eA, a$ and $b$ are two-sided ideals of $A$ and define a direct decomposition of $A$.