On Artinian and Noeterian modules relative to a torsion theory and ACC/DCC on essential submodules

The paper concern on the properties of Artinian and Noeterian modules relative to a torsion theory. It is shown that ACC (resp. DCC) on essential submodules equivalents that a module is Artinian (resp. Noeterian), for modules with finite Goldie dimension. It is shown that ACC (resp. DCC) on essential closed submodules equivalents factor on relative socle is Artinian (resp. Noeterian).