Exercise 2, § 1

(a) In the factorial polynomial domain $K[X, Y]$ in two indeterminates over a field $K$, give an example of two principal ideals $a, b$ such that $a + b$ is not divisorial.

(b) Let $K$ be the quadratic extension of the field $\mathbf{Q}(X)$ of rational functions in one indeterminate over $\mathbf{Q}$, obtained by adjoining to $\mathbf{Q}(X)$ a root $y$ of the polynomial $Y^2 - 2X \in \mathbf{Q}(X)[Y]$. Let $A$ be the subring of $K$ generated by $Z, X$ and $y$; $A$ is an integrally closed Noetherian domain. Show that $p = AX + Ay$ is a prime ideal of height 1 but that $p^2$ is not divisorial (show that $X^{-1}p^2$ is a prime ideal strictly containing $p$). Moreover $p . (A : p) \neq A$.

(c) Let $A$ be an integrally closed Noetherian local integral domain, whose maximal ideal $m$ is not of height $\leq 1$; let $p$ be a prime ideal of $A$ of height 1; for every integer $i > 0$ let $a_i = p + m^i$; show that $\operatorname{div}(\bigcap_i a_i)$ is distinct from $\sup(\operatorname{div}(a_i))$.