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A module [Formula: see text] is called a [Formula: see text]-module if, whenever [Formula: see text] and [Formula: see text] are submodules of [Formula: see text] with [Formula: see text] and [Formula: see text] is a homomorphism with [Formula: see text], we have [Formula: see text]. The class of [Formula: see text]-modules contains the [Formula: see text]-modules as well as the dual-square-free (DSF) modules. Furthermore, a [Formula: see text]-module [Formula: see text] is called pseudo-discrete if [Formula: see text] is also a lifting module. In this paper, we study the [Formula: see text]-, 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.

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