When some complement of an exact submodule is a direct summand

In this article, we study C11 r-modules. It is shown that the class of modules with property, each submodule such that the image of its quotient is zero under a left exact preradical has a complement which is a direct summand, is closed under direct sums. However, we provide examples which show that this new class of modules is not closed under direct summands. Failure of the latter closure property leads us to obtain several results on the inheritance of our condition by direct summands. We provide examples to illustrate our results by making special choices of left exact preradicals.