Torsion pairs over n-hereditary rings

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 Bézout ring which is 2-hereditary, but not 1-hereditary, such that the torsion pairs over this ring are not trivial.

Keywords:

• n-Hereditary
• n-coherent
• finitely n-presented
• FP_n-injective
• FP_n-flat
• torsion pairs
• Bézout ring
• injective cogenerator

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary 18E40
• Secondary 16E60

Acknowledgements

The authors wish to thank the referee for the careful reading and comments provided, which certainly have improved the quality of this article.

Additional information

Funding

Daniel Bravo was partially supported by CONICYT + FONDECYT/Regular +1180888. Carlos E. Parra was partially supported by CONICYT + FONDECYT/Iniciación + 11160078.