Exercise 22, § 1

(a) Let $\Gamma'$ be a lattice-ordered additive group and $A$ the algebra of $\Gamma$ over a field $k$; $A$ is an integral domain (Algebra, Chapter II, § 11, no. 4, Proposition 8). Every element $x \neq 0$ in $A$ may be written uniquely as

where the $v_i$ are distinct elements of $\Gamma$, the $e^{v_i}$ the corresponding elements of the canonical basis of $A$ over $k$ and the $a_i$ elements $\neq 0$ of $k$. Write $\phi(x) = \inf(v_1, \ldots, v_n)$ in $\Gamma$; show that $\phi(x+y) \geq \inf(\phi(x), \phi(y))$ if $x, y$ and $x+y$ are $\neq 0$ in $A$ and $\phi(xy) = \phi(x) + \phi(y)$ if $xy \neq 0$ in $A$. (To show the second assertion, establish first the following lemma: given a finite family $(\xi_j)$ of elements of $\Gamma$, in order that $\inf(\xi_j) = 0$, it is necessary and sufficient that for all $\eta > 0$ in $\Gamma'$ there exist an index $j$ and an element $\zeta \in \Gamma$ such that $0 < \zeta \leq \eta$ and $\inf(\xi_j, \zeta) = 0$. Apply this lemma reducing it to the case where $\phi(x) = \phi(y) = 0$.)

(b) Deduce from (a) that, if $K$ is the field of fractions of $A$, $\phi$ can be extended to a homomorphism from $K^*$ to $\Gamma$ (also denoted by $\phi$) such that

if $x, y$ and $x+y$ are $\neq 0$ in $K$. Deduce that, if $B$ is the set of $x \in K$ such that $x = 0$ or $\phi(x) \geq 0$, $B$ is a domain whose field of fractions is $K$ and such that, if $U$ is the group of invertible elements of $B$, $K^*/U$ is isomorphic to $\Gamma$ (and in particular $B$ is a pseudo-Bezoutian domain). Derive examples of a pseudo-Bezoutian domain which is not completely integrally closed, a completely integrally closed pseudo-Bezoutian domain which is not pseudo-principal (cf.

Algebra, Chapter VI, § 1, Exercise 31) \* and a pseudo-principal domain which is not factorial. \*

(c) Derive from (b) an example of an integral domain which is the union of a right directed family of factorial subdomains and is completely integrally closed but not pseudo-Bezoutian. (Let $\theta$ be an irrational number in $[0, 1]$ and for every integer $j$ let $q_j$ be the greatest integer such that $q_j / 2^j < 0$. For all $j$ define on the product group $\mathbf{Q} \times \mathbf{Q}$ a lattice-ordered group structure by taking the set $(\mathscr{G}_j)_+$ of elements $\geq 0$ of this group to be the ordered pairs $(\xi, \eta)$ such that $\xi \geq 0$ and $0 \leq \eta \leq (q_j 2^{-j}) \xi$. *)