C₁₂-modules via left exact preradicals

In this paper, we study modules with the condition that images of all submodules under a left exact preradical for the category of right modules over a ring can be essentially embedded in direct summands. This new class of modules properly contains the class of C₁₂-modules (and hence also CS-modules and uniform modules). It is shown that any module isomorphic to a direct summand of a module which satisfies the rC₁₂ property. In contrast to CS-modules, it is shown that the class of modules with the former property is closed under essential extensions whenever any module in the new class is relative injective with respect to its essential extensions.