Rings characterized by dual-Utumi-modules

Abstract

A right [Formula: see text]-module [Formula: see text] is called a dual-Utumi-module (DU-module) if for any two proper submodules [Formula: see text] and [Formula: see text] of [Formula: see text] with [Formula: see text] and [Formula: see text], there exist summands [Formula: see text] and [Formula: see text] of [Formula: see text] such that [Formula: see text], [Formula: see text] with [Formula: see text], [Formula: see text] small submodules of [Formula: see text] and [Formula: see text] a direct summand of [Formula: see text]. In this work, we characterize rings [Formula: see text] for which every (finitely generated, cyclic, free) [Formula: see text]-module is DU-module.