On strongly flat modules over integral domains (Q1880813)

A module \(C\) over a domain \(R\) is called weakly cotorsion if \(\text{Ext}^1_R(Q, C)=0\), where \(Q\) is the field of quotients of \(R\). An \(R\)-module \(M\) is strongly flat if \(\text{Ext}^1_R(M, C)=0\) for every weakly cotorsion module \(C\). The goal of this twenty-two page paper is to investigate strongly flat modules over general integral domains. Section 2 develops characterizations and properties of these modules over general domains with stronger results for Matlis domains. Section 3 gives more detailed information on strongly flat modules when \(R\) is a valuation domain, especially with regard to their pure submodules. One of the main results of the section, corollary 3.12, establishes that the class of strongly flat modules is closed under taking pure submodules of countable rank. Theorem 3.15 shows that every reduced strongly flat \(R\)-module of rank \(\aleph_1\) has a (free) dense basic submodule.