Abstract
A module M is called a D4-module if, whenever A and B are submodules of M with M = A circle plus B and f : A -> B is a homomorphism with Imf subset of(circle plus) B, we have kerf subset of(circle plus) A. The class of D4-modules contains the D3-modules as well as the dual-square-free (DSF) modules. Furthermore, a D4-module M is called pseudo-discrete if M is also a lifting module. In this paper, we study the D4-, the DSF, and the pseudo-discrete modules, and show that a pseudo-discrete module is clean iff it has the finite exchange property iff it has the full exchange property.