Exercise 30, § 1

An integral domain is called regularly integrally closed if, in the monoid D(A), every finitely generated divisor (Exercise 11) is a regular element. Every completely integrally closed domain is regularly integrally closed; every pseudo-Bezoutial domain (Exercise 2I) is regularly integrally closed. (a) If A is regularly integrally closed and $d, d', d''$ are three finitely generated divisors, the relation $d + d'' \leq d' + d''$ implies $d \leq d'$. (b) Deduce from (a) that, for an integral domain A to be regularly integrally closed, it is necessary and sufficient that, for every fractional ideal a of A such that div(a) is a finitely generated divisor, a: $a = A$; in particular A is integrally closed (Exercise6 (b)). (Use the fact that A: (bc) = (A: b) : c for two fractional ideals b, c of A, taking $b = a, c = A: a$.) Moreover, div(a) is then invertible in D(A).

(c) Let $A$ be a regularly integrally closed domain and $K$ its field of fractions; the monoid $D_f(A)$ of finitely generated divisors of $A$ generates in $D(A)$ a lattice-ordered group $G_f(A)$. For every polynomial $p \in K[X]$, let $d(p)$ denote the divisor of the fractional ideal of $A$ generated by the coefficients of $p$; for every rational function $r = p/q$ of $K(X)$, where $p, q$ are in $K[X]$ and $q \neq 0$, $d(p) - d(q)$ is well defined on $G_f(A)$ (independently of the expression of $r$ as a quotient of two polynomials); it is denoted by $d(r)$ (cf. Exercise 12 (c)). If we then write $\gamma(r) = ((r), d(r))$, where $(r)$ is the fractional principal ideal of $K[X]$ generated by $r$, $y$ is an isomorphism of the ordered group $\mathscr{P}^*(A[X])$ (notation of § 3, no. 2) onto a subgroup of the product ordered group

Deduce that $A[X]$ is a regularly integrally closed domain and that $D_f(A[X])$ is isomorphic to $\mathscr{P}^*(K[X]) \times D_f(A)$ (use (a) and (b) and Exercise 29 (b)).

(d) Let $B$ be an integrally closed domain, $K$ its field of fractions and $A$ the subring of the polynomial ring $K[X, Y]$ consisting of the polynomials whose constant term belongs to $B$. Show that, if $B \neq K$, $A$ is integrally closed but not regularly integrally closed (show that the divisor $\inf(\operatorname{div}(X), \operatorname{div}(Y))$ corresponds to a divisorial ideal $a$ such that $a : a \neq A$).