Dual Utumi modules.

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. [ABSTRACT FROM AUTHOR]