On the class group of a pullback (Q2785947)

Let \(R\) be an integral domain with quotient field \(K\). Let \(R:I=\{x\in K:xI\supset R\}\) and let \(I^{-1}=R:I\). Let \(I_v=(I^{-1})^{-1}\). If \(I=I_v\) then \(I\) is a \(v\)-ideal and \(I\) is \(v\)-finite if \(I=J_v\) for some finitely generated ideal \(J\), which is contained in \(I\). Define \(I_t=\bigcup\{J_v:0/J\supset I\), \(J\) is finitely generated\}. If \(I=I_t\), then \(I\) is a \(t\)-ideal. An integral domain \(R\) is PVMD (= ``Prüfer \(v\)-multiplication domain'') if the two equivalent conditions are satisfied:NEWLINENEWLINENEWLINE(i) The set of \(v\)-finite ideals forms a group, NEWLINENEWLINENEWLINE(ii) For each maximal \(t\)-ideal \(P\), \(R_P\) is a valuation domain.NEWLINENEWLINENEWLINELet \(T\) be an integral domain of the form \(K+M\), where \(K\) is a field and \(M\) is a maximal ideal of \(T\), and \(R=D+M\), where \(D\) is a subring of \(K\). \textit{D. F. Anderson} and \textit{A. Ryckaert} [J. Pure Appl. Algebra 52, No. 3, 199-212 (1988; Zbl 0668.13013)]\ calculated the class group and the local class group of \(R=D+M\) and gave necessary and sufficient conditions for \(R\) to be PVMD. The paper under review extends the above results proved for the \(D+M\) construction to a \(R=\pi^{-1}(D)\) constructed from the quasilocal domain \(T\) where \(\pi:T\to K\).NEWLINENEWLINEFor the entire collection see [Zbl 0855.00015].