Testing for cotorsionness over domains (Q957946)

Summary: We are looking for reduced modules over an integral domain to test the cotorsionness of modules. Our focus is on the cotorsion theories due to Matlis, Enochs and Warfield, respectively. If \(\kappa\) is a cardinal such that \(\kappa^{\aleph_0}=2^{\kappa}\), then for domains \(R\) for which \(Q/R\) (\(Q\) is the quotient field of \(R\)) is not self-small there is a reduced strongly flat \(R\)-module \(U_k\) of rank \(2^\kappa\) that can be used to test whether or not an \(R\)-module of cardinality \(2^k\) is Matlis cotorsion; however, no such module exists if the cardinality restriction is removed. We also establish the existence of reduced modules which test Matlis cotorsion modules for Enochs or Warfield cotorsionness. For a given torsion-free or flat \(R\)-module \(A\) of cardinality \(\kappa\), we construct an Enochs, resp. Matlis cotorsion \(R\)-module \(M\) of cardinality \(2^\kappa\) with \(\text{Ext}^1_R (A; M)=0\) that is not Warfield, resp. not Enochs cotorsion.