Characterization of Prüfer multiplication monoids and domains by means of spectral module systems. (Q1396720)

An integral domain \(A\) is called a Prüfer domain, resp. a Prüfer \(v\)-multiplication domain (PVMD), if every nonzero finitely generated ideal of \(A\) is invertible, resp. every finite type \(t\)-ideal is \(t\)-invertible. It is well-known that there exist several characterizations of the Prüfer domains by properties of their overrings (overring = ring between \(A\) and its quotient field). Most of them have \(t\)-analogs for PVMDs. The aim of the paper under review is to generalize/unify such results in the context of commutative cancellative monoids with zero. Let \(D\) be such a monoid and \(r\) a finitary ideal system on \(D\) (e.g. \(r\) = the \(t\)-system). \(D\) is called an \(r\)-Prüfer monoid if every non-zero \(r\)-finitely generated \(r\)-ideal is \(r\)-invertible. An overmonoid \(T\) of \(D\) is a monoid between \(D\) and its quotient groupoid. \(T\) is said to be \(r\)-linked if \((M\cap D)_r\neq D\) for each \(t\)-maximal \(t\)-ideal \(M\) of \(T\). And \(T\) is said to be pseudo-flat over \(D\) if \(T_M=D_{M\cap D}\) for each \(t\)-maximal \(t\)-ideal \(M\) of \(T\). One of the main results of the paper asserts that a monoid \(D\) is \(r\)-Prüfer iff \(D\) is root-closed and every \(r\)-linked overmonoid of \(D\) is an intersection of quotient monoids of \(D\) iff \(D\) is root-closed and every \(r\)-linked overmonoid of \(D\) is seminormal iff every \(r\)-linked overmonoid of \(D\) is pseudo-flat over \(D\).