On the generalization of torsion functor and P-semiprime modules over noncommutative rings

Let R be an associative Noetherian unital noncommutative ring R. We introduce the functor P Gamma;P over the category of R-modules and use it to characterize P-semiprime. P-semisecond R-modules also characterized by the functor P Lambda;P. We also show that the Greenless-May Duality (GM) and Matlis Greenless-May Equality(MGM) hold over the full subcategory of R-Mod consisting of R-semiprime and R-semisecond modules. Finally, we generate a one-sided right ideal PTP(R), which gives an equivalent formulation to solve Köthe conjecture positively or negatively.