Modules that have a supplement in every coatomic extension

Abstract

Let R be a ring and M be an R-module. M is said to be an E-module (respectively, an EE-module) if M has a supplement (respectively, ample supplements) in every coatomic extension N , i.e. N M is coatomic. We prove that if a module M is an EE-module, every submodule of M is an E-module, and then we show that a ring R is left perfect iff every left R-module is an E-module iff every left R-module is an EE-module. We also prove that the class of E-modules is closed under extension. In addition, we give a new characterization of left V-rings by cofinitely injective modules.