Rings whose cyclics are U-modules

A right R-module M is called a U-module if, whenever A and B are submodules of M with A congruent to B and A boolean AND B = 0, there exist two direct summands K and T of M such that A subset of(ess) K, B subset of(ess) T and K circle plus T subset of(circle plus) M. The class of U-modules is a strict and simultaneous generalization of quasi-continuous, square-free and automorphism-invariant modules. In this paper we study the rings whose cyclics are U-modules, extending many of the known results on the subject and providing new ones.