Exercise 19, § 9

Let A be a ring, $\mathfrak{a}$ a two-sided ideal of A contained in $\mathfrak{R}$(A), let $e$ and $e'$ be idempotents in A, and let $e$ and $e'$ their classes in the ring $A = A/\mathfrak{a}$. Prove that if the idempotents $e$ and $e'$ are equivalent (VIII, p. 39, Exercise 5), then so are $e$ and $e'$ (observe that $Ae$ is a projective cover of the A-module $Ae$).