Gradual and Fuzzy Modules: Functor Categories

The categorical treatment of fuzzy modules presents some problems, due to the well known fact that category is not abelian, and even normal. Our aim give a representation inside generalized modules, in fact, functor category, Mod−P, which Grothendieck category. To do that, first we consider preadditive P, defined by interval P=(0,1], build torsionfree class J hereditary torsion theory finally identify equivalence classes submodules module M with F-pair, are pair (G,F), decreasing gradual M, where G belongs J, satisfying G=Fd, ∪αF(α) disjoint union F(1) F(α)\G(α), α running (0,1].