Rings whose RD-flat modules have restricted subflat domains

A module K-R is said to be L-R-subflat if for every short exact sequence 0 ? U? D? L? 0 of left R-modules, the sequence 0 ? K? U? K? D? K? L? 0 is exact. The subflat domains of (RD-flat) modules somehow tells us how far (or how close) such a module is from being flat. Every right R-module is subflat relative to all flat left R-modules, and flat modules are the only ones sharing the distinction of being in every single subflat domain. A module is called f-test if it is subflat only to flat modules. Similarly, an RD-flat module is called tf-test if it is subflat only to torsion-free modules. In this paper, we consider two families of rings characterized by their RD-flat modules: those whose finitely presented RD-flat modules are either flat or tf-test (property (P)) and those whose finitely presented RD-flat modules are either torsion-free (flat) or f-test (property (Q)). Structural properties of both classes of rings are studied.