On torsion-free modules and semi-hereditary rings

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.