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