# Torsion pairs over $n$-Hereditary rings

## TL;DR

This paper explores the properties of $n$-hereditary rings, characterizes them via module quotients and submodules, and examines torsion pairs related to FP$_n$-injective and FP$_n$-flat modules, including a non-trivial example.

## Contribution

It introduces new characterizations of $n$-hereditary and $n$-coherent rings and analyzes torsion pairs over these rings, providing a novel example of a non-trivial case.

## Key findings

- Characterization of $n$-hereditary rings via injective and flat modules
- Existence of torsion pairs related to FP$_n$-injective and FP$_n$-flat modules
- Example of a 2-hereditary Bézout ring with non-trivial torsion pairs

## Abstract

We study the notions of $n$-hereditary rings and its connection to the classes of finitely $n$-presented modules, FP$_n$-injective modules, FP$_n$-flat modules and $n$-coherent rings. We give characterizations of $n$-hereditary rings in terms of quotients of injective modules and submodules of flat modules, and a characterization of $n$-coherent using an injective cogenerator of the category of modules. We show two torsion pairs with respect to the FP$_n$-injective modules and the FP$_n$-flat modules over $n$-hereditary rings. We also provide an example of a B\'ezout ring which is 2-hereditary, but not 1-hereditary, such that the torsion pairs over this ring are not trivial.