RINGS WHOSE CYCLIC MODULES ARE RADICAL LIFTING MODULES

A right R-module M is called a radical lifting if for every submodule N of M, with Rad (M) <= N, there exists a module decomposition M = M-1 circle plus M-2 such that M-1 <= N and N boolean AND M-2 is small in M-2. 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.