On Semisimple-direct Modules

The direct-injective module and its generalizations have been studied in several articles. Recently, semisimple versions of direct injective modules have been studied. Namely, a module is referred to as a semisimple-direct injective provided that for any two isomorphic semisimple submodules of A, one of them is a direct summand of A, then the same is true for the other. A study on various fundamental properties of these modules has been done, which enables us to obtain some new results on semisimple-direct injective modules. We also use semisimple-direct injective modules to characterize few well-known rings.