On torsion-free modules and semi-hereditary rings

Comment

Semi-hereditary rings form an important class in commutative algebra when the Noetherian hypothesis is not assumed—for example in the study of perfectoid rings. We prove several results on the structure of this class, focusing on the relationship between semi-hereditary rings and flatness of torsion-free modules. We also discuss Shimomoto’s problem on flatness of the Frobenius map.