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.