Building pretorsion theories from torsion theories

TL;DR

This paper explores how pretorsion theories, generalizations of torsion theories, naturally occur in abelian categories and introduces methods to derive pretorsion theories from torsion theories, with various applications.

Contribution

It demonstrates that pretorsion theories can be constructed from torsion theories in abelian categories using two distinct approaches and provides a universal method in additive categories.

Findings

Pretorsion theories appear in classical abelian categories.

Two methods to derive pretorsion theories from torsion theories are proposed.

Applications include module categories, internal groupoids, and representation theory.

Abstract

Torsion theories play an important role in abelian categories and they have been widely studied in the last sixty years. In recent years, with the introduction of pretorsion theories, the definition has been extended to general (non-pointed) categories. Many examples have been investigated in several different contexts, such as topological spaces and topological groups, internal preorders, preordered groups, toposes, V-groups, crossed modules, etc. In this paper, we show that pretorsion theories naturally appear also in the "classical" framework, namely in abelian categories. We propose two ways of obtaining pretorsion theories starting from torsion theories. The first one uses "comparable" torsion theories, while the second one extends a torsion theory with a Serre subcategory. We also give a universal way of obtaining a torsion theory from a given pretorsion theory in additive categories. We conclude by providing several applications in module categories, internal groupoids, recollements and representation theory.