Abstract
A submodule N of a module M is called $$\mathcal {D}$$-closed if the socle of M / N is zero. $$\mathcal {D}$$-closed submodules are similar to $$\mathcal {S}$$-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 by $$\mathcal {D}$$-closed submodules in terms of that submodule, and show that 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.
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