On modules which are isomorphic to relatively divisible or pure submodules of each other

Abstract

In this note, we provide a generalization of a well-known result of module theory which states that two injective modules are isomorphic when they are isomorphic to submodules of each other. More precisely, we show here that two RD-injective (respectively, pure-injective) modules over an integral domain are isomorphic if they are isomorphic to relatively divisible (respectively, pure) sub-modules of each other.

Keywords: Relatively divisible submodules, pure submodules, injective modules, RD- injective modules, pure-injective modules