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.

Model structures, recollements and duality pairs

In this paper, we construct some model structures corresponding Gorenstein [Formula: see text]-modules and relative Gorenstein flat modules associated to duality pairs, Frobenius pairs and cotorsion pairs. By investigating homological properties of Gorenstein [Formula: see text]-modules and some known complete hereditary cotorsion pairs, we describe several types of complexes and obtain some characterizations of Iwanaga–Gorenstein rings. Based on some facts given in this paper, we find new duality pairs and show that [Formula: see text] is covering as well as enveloping and [Formula: see text] is preenveloping under certain conditions, where [Formula: see text] denotes the class of Gorenstein [Formula: see text]-injective modules and [Formula: see text] denotes the class of Gorenstein [Formula: see text]-flat modules. We give some recollements via projective cotorsion pair [Formula: see text] cogenerated by a set, where [Formula: see text] denotes the class of Gorenstein [Formula: see text]-projective modules. Also, many recollements are immediately displayed through setting specific complete duality pairs.

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.

Gorenstein flat modules relative to injectively resolving subcategories

Let [Formula: see text] be an injectively resolving subcategory of left [Formula: see text]-modules. We introduce and study [Formula: see text]-Gorenstein flat modules as a common generalization of some known modules such as Gorenstein flat modules (Enochs, Jenda and Torrecillas, 1993), Gorenstein AC-flat modules (Bravo, Estrada and Iacob, 2018). Then we define a resolution dimension relative to the [Formula: see text]-Gorensteinflat modules, investigate the properties of the homological dimension and unify some important properties possessed by some known homological dimensions. In addition, stability of the category of [Formula: see text]-Gorensteinflat modules is discussed, and some known results are obtained as applications.