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, FPn-injective modules, FPn-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 FPn-injective modules and the FPn-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.

Torsion and torsion-free classes from objects of finite type in Grothendieck categories

“…Applications of the mentioned characterizations of I n and F n as torsion and torsion-free classes, respectively, are given within the contexts of modules, chain complexes, and functor categories, widely used in module theory and representation theory of Artin algebras. Thus, we recover some known results, such as characterizations of n-hereditary rings (see [6]), but more important, we also obtain several interesting outcomes in the setting of functors categories, like for instance a description of semi-hereditary rings R in terms of solutions of linear systems over R.…”

“…We only prove the implication (iii⇒i). We essentially follow the argument from [2,6]. For the reader's convenience, we provide a proof for more general Grothendieck categories.…”

“…As we have proved in Theorem 3.14, whenever the category G has a generating set of small projective objects, being F n (G) a torsion-free class and I n (A) a torsion class are equivalent for n ≥ 2. When G := R op -Mod, it is equivalent to being R left n-hereditary; see [6,Thms. 5.3 and 5.5].…”

“…Recall from [6,Def. 7] that a ring R is left n-hereditary, n ≥ 0, if every submodule of type FP n−1 3 of a finitely generated projective left R-module is projective, too.…”

“…In general, being R a left n-hereditary ring is characterized with pd(FP n (R)) ≤ 1 (see [6,Lem. 8]) which leads us to make the following.…”