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 and 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.