On Representable Rings and Modules

In this paper, we determine the structure of rings that have secondary repre-sentation (called representable rings) and give some characterizations of these rings. Also, we characterize Artinian rings in terms of representable rings. Then we introduce com-pletely representable modules, modules every non-zero submodule of which have secondary representation, and give some necessary and sufficient conditions for a module to be com-pletely representable. Finally, we define strongly representable modules and give some conditions under which representable module is strongly representable.