Exercise 15, § 4

Let $A$ be a ring and $a$ a finitely generated ideal of $A$. Show that the following properties are equivalent: ($a$) $a^2 = a$; ($\beta$) $A$ is generated by an idempotent; (y) $V(a)$ is both open and closed in $X = \operatorname{Spec}(A)$ and $a$ is minimal among the ideals $b$ such that $r(b) = r(a)$ (use § 2, Corollary 3 to Proposition 4). Give an example where $a^2 = a$, $a$ is not finitely generated and does not contain an idempotent (cf. § 2, Exercise 15).