On D-closed submodules

A submodule N of a module M is called D-closed if the socle of M/ N is zero. D-closed submodules are similar to S-closed submodules (a generalization of closed submodules) defined through nonsingular modules. First, we describe the smallest proper class (due to Buchsbaum) containing the class of short exact sequences determined byD-closed submodules in terms of that submodule, and showthat it coincides with other classes of modules under certain conditions. Second, we study coprojective modules of this class, called edc-flat modules. We give some equivalent conditions for injective modules to be edc-flat for special rings, and for edc-flat modules to be projective (flat) for any ring