Linear filters and hereditary 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.

Gradual and Fuzzy Modules: Functor Categories

“…Observe that this Gabriel filter L can be also described as the filter of all ideals a ⊆ Hom P (a, −) satisfying that for every b ∈ P, and any g ∈ Hom P (a, b), there exist c ∈ P, and h ∈ Hom P (b, c) such that hg ∈ a(c). These are the dense ideals in [7].…”

“…In [7], a correspondence is established between Gabriel filters and hereditary torsion theories that associates to any Gabriel filter L = {L(a) | a ∈ P}; the torsion class T (L):…”