Exercise 19, § 2

An integral domain $A$ is called pseudo-Priiferian if the set of finitely generated divisors of $D(A)$ ($§ 1$, Exercise 11) is a group. A pseudo-Priiferian domain is regularly integrally closed ($§ 1$, Exercise 29). A pseudo-Bezoutian domain ($§ 1$, Exercise 21) is pseudo-Priiferian; a Priiferian domain is pseudo-Priiferian; a Krull domain is pseudo-Priiferian.

(a) Show by examples that the converses of the last three assertions are not necessarily true.

(b) Let $\theta$ be an irrational number $> 0$ and $\Gamma$ the group $\mathbf{R}^2$ ordered by taking the set of ordered pairs $(a, \beta)$ such that $a \geq 0$ and $\beta \geq \theta a$ as the set of positive elements; the ordered group $\Gamma$ is a complete lattice. Let $B$ be the pseudo-principal domain derived from $\Gamma$ by the procedure of $§ 1$, Exercise 22 and let $A$ be the subring the intersection of $B$ and the algebra over $k$ of the subgroup $\mathbf{Q}^2$ of $\mathbf{R}^2$. Show that $A$ is a completely integrally closed domain but is not pseudo-Priiferian (note that, if $K$ is the field of fractions of $A$ and $U$ the group of invertible elements of $A$, the ordered group $K^*/U$ is isomorphic to the group $\Gamma \cap \mathbf{Q}^2$, ordered by the ordering induced by that on $\Gamma$).

(c) If the domain $\mathbf{A}$ is pseudo-Pruferian, show that the polynomial domain $\mathbf{A}[X]$ is pseudo-Pruferian (cf. § 1, Exercise 30 (c)).

(d) Let $\mathbf{A}$ be a pseudo-Pruferian domain and $\mathbf{K}$ its field of fractions. Show that the integral closure $\mathbf{B}$ of $\mathbf{A}$ in an algebraic extension $\mathbf{L}$ of $\mathbf{K}$ is a pseudo-Pruferian domain (method of Exercise 16, using § 1, Exercise 29 (a)).

(e) Deduce from Exercise 30 (d) of § 1 an example of an increasing sequence $(\mathbf{A}_i)$ of factorial domains with the same field of fractions whose union is not a regularly integrally closed domain (nor a fortiori a pseudo-Pruferian domain) (in the example quoted, take $\mathbf{B}$ to be a discrete valuation ring). \*