Abstract
A submodule N of a right R-module M is said to lie over a direct summand of M if, there is a decomposition
with
and
. In this paper, a right R-module M is called a Dual-Utumi-Module (
-module) if for any two proper submodules A and B of M with
and A + B = M, there exist two summands K and L of M such that A lies over K, B lies over L and
Dual-U-modules are strict generalizations of quasi-discrete, pseudo-discrete, and dual-square-free modules. In this paper several characterizations of
-modules are provided which in turn are used to show that the class of
-modules inherits some of the important features of the aforementioned classes of modules. For example, if
is a
-module with
then M is quasi-projective and discrete. As an immediate application, a ring R is right perfect iff every quasi-projective right R-module is a
-module. It is also shown that if M is a
-module whose local summands are summands, then
where Q and P are factor-orthogonal, Q is P-projective and dual-square-free,
is a direct sum of pairwise non-isomorphic indecomposable modules,
is quasi-projective and discrete with
, and C is isomorphic to a direct summand of
. In particular, every quasi-discrete module has such a decomposition.