Modules that have a supplement in every cofinite extension

Abstract

Let R be a ring and M a left R-module. An R-module N is called a cofinite extension of M in case M subset of N and N/M is finitely generated. We say that M has the property (CE) (resp. (CEE)) if M has a supplement (resp. ample supplements) in every cofinite extension. In this study we give various properties of modules with these properties. We show that a module M has the property (CEE) iff every submodule of M has the property (CE). A ring R is semiperfect iff every left R-module has the property (CE). We also study cofinitely injective modules, direct summands of every cofinite extension, as a generalization of injective modules.