Exercise 1, § 1

(a) Let $A$ be an absolutely flat ring (Chapter I, §2, Exercise 17) Show that $\operatorname{Ass}(A)$ is the set of isolated points of $\operatorname{Spec}(A)$ (note that $\operatorname{Ass}(A)$ is the set of prime ideals which are annihilators of an idempotent of $A$).

(b) Deduce from (a) an example of a ring $A$ such that $A \neq 0$ and $\operatorname{Ass}(A) = \varnothing$ (cf. Chapter II, §4, Exercise 17) and for which the conclusion of no. 1, Corollary 2 to Proposition 2 does not hold.

(c) Deduce from (b) an example of a ring $A$ and a multiplicative subset $S$ of $A$ such that the mapping $p \mapsto S^{-1}p$ of $\operatorname{Ass}(A) \cap \Phi$ to $\operatorname{Ass}(S^{-1}A)$ (no. 2, Proposition 5) is not surjective.