Abstract
A right R-module M is called a U-module if, whenever A and B are submodules of M with A≅B and A ∩ B = 0, there exist two summands K and L of M such that A⊆
ess
K, B⊆
ess
L and K⊕L⊆
⊕
M. The class of U-modules is a simultaneous and strict generalization of three fundamental classes of modules; namely, the quasi-continuous, the square-free, and the automorphism-invariant modules. In this paper we show that the class of U-modules inherits some of the important features of the aforementioned classes of modules. For example, a U-module M is clean if and only if it has the finite exchange property, if and only if it has the full exchange property. As an immediate consequence, every strongly clean U-module has the substitution property and hence is Dedekind-finite. In particular, the endomorphism ring of a strongly clean U-module has stable range 1.