Some polynomial characterizations of Prüfer v-multiplication domains (Q794711)

Let D be a commutative integral domain. For each non-zero (fractional) ideal A of D, let \(A_ v=(A^{-1})^{-1}\). The ideal B is said to be a v-ideal of finite type if \(B=A_ v\) for some finitely generated A. The set H(D) of such ideals has a multiplication given by \(B.C=B_ vC_ v\), and D is said to be a Prüfer v-multiplication domain if H(D) is then a group. Given a polynomial f in D[X], let \(A_ f\) be the ideal of D generated by the coefficients of f. Let S be the set of polynomials with \((A_ f)_ v=D\). The author's main result is that the following are equivalent when D is integrally closed: (i) D is a Prüfer v-multiplication domain, (ii) \(D[X]_ s\) is Prüfer, (iii) \(D[X]_ s\) is Bezout.