w -locally isomorphic torsionless modules over weakly Matlis domains

Let R be an integral domain. An R-module G is torsionless if it is isomorphic to a submodule of a finitely generated free R-module. For torsionless modules, we define w-weak isomorphism types which are new versions of weak isomorphism types with respect to the w-operation: w-local isomorphism and w-nearly isomorphism. Furthermore, we examine the relationship between w-local isomorphism, w-nearly isomorphism, and stable isomorphism for torsionless w-modules over weakly Matlis domains.