The following is Exercise 2.4, in Chapter 1 of Liu, Algebraic Geometry and Arithmetic Curves:
Let $I$ be a finitely generated ideal of $A$:
$A/I$ is flat.
$I^2 = I$.
$I = (e)$ where $e^2=e$.
I can show that $2\iff 3$ and that $1 \implies 2$, and I remember proving the other way before but cannot recall it now. That is, I would like to show that $A/I$ is flat assuming that it is principal and generated by an idempotent.
I am sure this is proved somewhere on here, but i could not find the link. But here it is: $A=(e)\times (1-e)$, so $(1-e)\cong A/I$ is projective.