New characterizations of S-coherent rings

In this paper, we introduce and study the class [Formula: see text]-[Formula: see text]-ML of [Formula: see text]-Mittag-Leffler modules with respect to all flat modules. We show that a ring [Formula: see text] is [Formula: see text]-coherent if and only if every ideal is in [Formula: see text]-[Formula: see text]-ML, if and only if [Formula: see text]-[Formula: see text]-ML is closed under submodules. As an application, we obtain the [Formula: see text]-version of Chase Theorem: a ring [Formula: see text] is [Formula: see text]-coherent if and only if any direct product of copies of [Formula: see text] is [Formula: see text]-flat, if and only if any direct product of flat [Formula: see text]-modules is [Formula: see text]-flat. Consequently, we provide an answer to the open question proposed by Bennis and El Hajoui [On [Formula: see text]-coherence, J. Korean Math. Soc. 55(6) (2018) 1499–1512].

Gorenstein n-X -Injective and n - X-Flat Modules with Respect to a Special Finitely Presented Module

Let [Formula: see text] be a ring, [Formula: see text] a class of [Formula: see text]-modules and [Formula: see text] an integer. We introduce the concepts of Gorenstein [Formula: see text]-[Formula: see text]-injective and [Formula: see text]-[Formula: see text]-flat modules via special finitely presented modules. Besides, we obtain some equivalent properties of these modules on [Formula: see text]-[Formula: see text]-coherent rings. Then we investigate the relations among Gorenstein [Formula: see text]-[Formula: see text]-injective, [Formula: see text]-[Formula: see text]-flat, injective and flat modules on [Formula: see text]-[Formula: see text]-rings (i.e., self [Formula: see text]-[Formula: see text]-injective and [Formula: see text]-[Formula: see text]-coherent rings). Several known results are generalized to this new context.

K-theory of n-coherent rings

Let [Formula: see text] be a strong [Formula: see text]-coherent ring such that each finitely [Formula: see text]-presented [Formula: see text]-module has finite projective dimension. We consider [Formula: see text] the full subcategory of [Formula: see text]-Mod of finitely [Formula: see text]-presented modules. We prove that [Formula: see text] is an exact category, [Formula: see text] for every [Formula: see text] and we obtain an expression of [Formula: see text].

Some remarks on Nonnil-coherent rings and ϕ-IF rings

Let [Formula: see text] be a commutative ring. If the nilpotent radical [Formula: see text] of [Formula: see text] is a divided prime ideal, then [Formula: see text] is called a [Formula: see text]-ring. In this paper, we first distinguish the classes of nonnil-coherent rings and [Formula: see text]-coherent rings introduced by Bacem and Ali [Nonnil-coherent rings, Beitr. Algebra Geom. 57(2) (2016) 297–305], and then characterize nonnil-coherent rings in terms of [Formula: see text]-flat modules, nonnil-injective modules and nonnil-FP-injective modules. A [Formula: see text]-ring [Formula: see text] is called a [Formula: see text]-IF ring if any nonnil-injective module is [Formula: see text]-flat. We obtain some module-theoretic characterizations of [Formula: see text]-IF rings. Two examples are given to distinguish [Formula: see text]-IF rings and IF [Formula: see text]-rings.

Relative coherent modules

Several authors have introduced various types of coherent-like rings and proved analogous results on these rings. It appears that all these relative coherent rings and all the used techniques can be unified. In [Formula: see text]-[Formula: see text]-coherent rings, Int. Electron. J. Algebra, 7 (2010) 128–139, several coherent-like rings are unified. In this paper, we continue this work and we introduce coherent-like module which also emphasizes our point of view by unifying the existed relative coherent concepts. Several classical results are generalized and some new results are given.

On adjoint resolutions and dimensions of modules

We introduce and investigate the adjoint resolutions and adjoint dimensions of modules. As a consequence, we give some new characterizations of weak global dimensions of coherent rings in terms of adjoint resolutions and adjoint dimensions of modules.