A right R-module M is called a radical lifting if for every submodule N of M, with there exists a module decomposition  such that  and  is small in  In this paper, we study sufficient conditions for a direct summand, factor module of a radical lifting module to be radical lifting. Thereafter, we study rings over which every cyclic right R-module  is a radical lifting module. Examples are provided to illustrate the necessity and the sufficiency of the conditions in our result. We also provide examples to show that the class of rings all of whose cyclic right modules are radical lifting is not left-right symmetric.

supplemented modules, H-supplemented modules, ⊕-supplemented modules, lifting modules, radical lifting modules, semiperfect rings.