On Rad-D12 Modules

Let M be a right R-module. We call M Rad-D12, if for every sub- module N of M, there exist a direct summand K of M and an epimor- phism α : K → M/N such that Ker_erα ⊆ Rad(K). We show that a direct summand of a Rad-D₁₂ module need not be a Rad-D12 module. We investigate completely Rad-D₁₂ modules (modules for which every direct summand is a Rad-D12 module). We also show that a direct sum of Rad-D12 modules need not be a Rad-D12 module. Then we deal with some cases of direct sums of Rad-D12 modules.