Let M be a right R-module. We call M Rad-D-12, if for every submodule N of M, there exist a direct summand K of M and an epimorphism alpha : K -> M/N such that Ker alpha subset of Rad(K). We show that a direct summand of a Rad-D-12 module need not be a Rad-D-12 module. We investigate completely Rad-D-12 modules (modules for which every direct summand is a Rad-D-12 module). We also show that a direct sum of Rad-D-12 modules need not be a Rad-D-12 module. Then we deal with some cases of direct sums of Rad-D-12 modules.