On torsion-free modules and semi-hereditary rings

TL;DR

This paper explores the structure of semi-hereditary rings and their relation to torsion-free modules' flatness, addressing key questions in non-Noetherian ring theory and the flatness of the Frobenius map.

Contribution

It provides new results on the structure of semi-hereditary rings and their connection to torsion-free modules, including insights into Shimomoto's problem.

Findings

Established structural properties of semi-hereditary rings

Linked flatness of torsion-free modules to ring properties

Addressed flatness of the Frobenius map in specific contexts

Abstract

The class of semi-hereditary rings is an important class of rings in theories that do not assume the Noetherian condition, such as perfectoid ring theory. We prove several results concerning the structure theory of this class, focusing on the relationship between semi-hereditary rings and the flatness of torsion-free modules. We also consider Shimomoto's problem concerning the flatness of the Frobenius map.