Gradual and Fuzzy Modules: Functor Categories García Hernández, Josefa María Jara Martínez, Pascual Fuzzy set Fuzzy module Gradual element Gradual module Gradual ring Functorial category The categorical treatment of fuzzy modules presents some problems, due to the well known fact that the category of fuzzy modules is not abelian, and even not normal. Our aim is to give a representation of the category of fuzzy modules inside a generalized category of modules, in fact, a functor category, Mod-P, which is a Grothendieck category. To do that, first we consider the preadditive category P, defined by the interval P = (0,1], to build a torsionfree class J in Mod-P, and a hereditary torsion theory in Mod-P, to finally identify equivalence classes of fuzzy submodules of a module M with F-pair, which are pair (G, F), of decreasing gradual submodules of M, where G belongs to J, satisfying G = F-d, and U alpha F (alpha) is a disjoint union of F(1) and F(alpha)\G(alpha), where alpha is running in (0, 1]. 2022-12-13T12:30:09Z 2022-12-13T12:30:09Z 2022-11-15 info:eu-repo/semantics/article García, J.M.; Jara, P. Gradual and Fuzzy Modules: Functor Categories. Mathematics 2022, 10, 4272. [https://doi.org/10.3390/math10224272] https://hdl.handle.net/10481/78426 10.3390/math10224272 eng http://creativecommons.org/licenses/by/4.0/ info:eu-repo/semantics/openAccess Atribución 4.0 Internacional MDPI