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.

18 pages. Added new results as Theorem 24 and Theorem 26 which characterize n-hereditary and n-coherent rings, respectively. Also added Appendix B which gives a module theoretic result obtained in Grothendieck categories