Exercise 2, § 1

Let $K$ be a commutative field and $A$ the valuation ring, whose order group is $\mathbf{Q}$, consisting of the "formal power series" $\sum_r c_r T^r$, where $c_r \in K$, $r \in \mathbf{Q}_+$ and the set of $r$ such that $c_r \neq 0$ is well ordered. Let $a$ be an irrational number $> 0$, $a$ the ideal of $A$ consisting of the elements with valuation $> a$ and $C$ the ring $A/a$. Then $\operatorname{Spec}(C)$ reduces to a point $p$; for all $x \neq 0$ in $C$, $px \neq 0$, but for all $\lambda \in p$, there exists $y \neq 0$ in $C$ such that $\lambda y = 0$. In particular, $\operatorname{Ass}(C) = \varnothing$, although $\operatorname{Supp}(C) = \operatorname{Spec}(C) = \{p\}$.