Llinear Filters and Hereditary Torsion Theories in Functor Categories

TL;DR

This paper generalizes the correspondence between Gabriel filters and hereditary torsion theories from rings to preadditive categories, expanding the theoretical framework in functor categories.

Contribution

It introduces Gabriel filters for preadditive categories and establishes a bijective correspondence with hereditary torsion theories in functor categories, generalizing classical results.

Findings

Established a bijective correspondence between Gabriel filters and hereditary torsion theories in (C,Ab)

Generalized classical theorems from rings to preadditive categories

Extended the theoretical framework of torsion theories in functor categories

Abstract

We introduce the notion of Gabriel filter for a preadditive category C and we show that there is a bijective correspondence between Gabriel filters of C and hereditary torsion theories in the category of additive functors (C,Ab), obtaining a generelization of the theorem given by Gabriel [Ga] and Maranda [Ma] which establishes a bijective correspondence between Gabriel filters for a ring and hereditary torsion theories in the corresponding category of modules.