Absolutely supplement and absolutely complement modules

Abstract

We introduce and study absolutely supplement (respectively complement) modules. We call a module an absolutely supplement (respectively complement) if it is a supplement (respectively complement) in every module containing it. We show that a module is absolutely supplement (respectively complement) if and only if it is a supplement (respectively complement) in its injective envelope. The class of all absolutely supplement (respectively complement) modules is closed under extensions and under supplement submodules (respectively under factor modules by complement submodules). We also consider the dual notions of absolutely co-supplements (respectively co-complements).