1705.03840May 10, 201718 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
We study the notions of n-hereditary rings and its connection to the classes of finitely n-presented modules, FPn-injective modules, FPn
Nearby in the stack
-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.