It is known that a ring $R$ is left Noetherian if and only if every left $R$-module has an injective (pre)cover. We show that $\left(1\right)$ if $R$ is a right $n$-coherent ring, then every right $R$-module has an $(n,d)$-injective (pre)cover; $\left(2\right)$ if $R$ is a ring such that every $(n,0)$-injective right $R$-module is $n$-pure extending, and if every right $R$-module has an $(n,0)$-injective cover, then $R$ is right $n$-coherent. As applications of these results, we give some characterizations of $(n,d)$-rings, von Neumann regular rings and semisimple rings....

Let $R$ be an associative ring with identity and $ℱ$ a class of $R$-modules. In this article: we first give a detailed treatment of Cartan-Eilenberg $ℱ$ complexes and extend the basic properties of the class $ℱ$ to the class $\mathrm{CE}(ℱ$). Secondly, we study and give some equivalent characterizations of Cartan-Eilenberg projective, injective and flat complexes which are similar to projective, injective and flat modules, respectively. As applications, we characterize some classical rings in terms of these complexes,...

Let $R$ be a commutative ring and $𝒞$ a semidualizing $R$-module. We investigate the relations between $𝒞$-flat modules and $𝒞$-FP-injective modules and use these modules and their character modules to characterize some rings, including artinian, noetherian and coherent rings.

Let $𝒜$ and $ℬ$ be abelian categories with enough projective and injective objects, and $T:𝒜\to ℬ$ a left exact additive functor. Then one has a comma category $\left(ℬ\downarrow T\right)$. It is shown that if $T:𝒜\to ℬ$ is $𝒳$-exact, then ${(}^{\perp }𝒳,𝒳)$ is a (hereditary) cotorsion pair in $𝒜$ and ${(}^{\perp }𝒴,𝒴)$) is a (hereditary) cotorsion pair in $ℬ$ if and only if $\left(\left(\genfrac{}{}{0pt}{}{{}^{\perp }𝒳}{{}^{\perp }𝒴}\right)\right),\langle 𝐡(𝒳,𝒴)\rangle )$ is a (hereditary) cotorsion pair in $\left(ℬ\downarrow T\right)$ and $𝒳$ and $𝒴$ are closed under extensions. Furthermore, we characterize when special preenveloping classes in abelian categories $𝒜$ and $ℬ$ can induce special preenveloping classes...

Existence of proper Gorenstein projective resolutions and Tate cohomology is proved over rings with a dualizing complex. The proofs are based on Bousfield Localization which is originally a method from algebraic topology.

Let $(𝒜,ℬ)$ be a complete and hereditary cotorsion pair in the category of left $R$-modules. In this paper, the so-called Gorenstein projective complexes with respect to the cotorsion pair $(𝒜,ℬ)$ are introduced. We show that these complexes are just the complexes of Gorenstein projective modules with respect to the cotorsion pair $(𝒜,ℬ)$. As an application, we prove that both the Gorenstein projective modules with respect to cotorsion pairs and the Gorenstein projective complexes with respect to cotorsion pairs possess...

We give a survey of our recent results on homological properties of Köthe algebras, with an emphasis on biprojectivity, biflatness, and homological dimension. Some new results on the approximate contractibility of Köthe algebras are also presented.

Let $R$ be a graded ring and $n\ge 1$ an integer. We introduce and study $n$-strongly Gorenstein gr-projective, gr-injective and gr-flat modules. Some examples are given to show that $n$-strongly Gorenstein gr-injective (gr-projective, gr-flat, respectively) modules need not be $m$-strongly Gorenstein gr-injective (gr-projective, gr-flat, respectively) modules whenever $n>m$. Many properties of the $n$-strongly Gorenstein gr-injective and gr-flat modules are discussed, some known results are generalized. Then we investigate...

We consider functorially finite subcategories in module categories over Artin algebras. One main result provides a method, in the setup of bounded derived categories, to compute approximations and the end terms of relative Auslander-Reiten sequences. We also prove an Auslander-Reiten formula for the setting of functorially finite subcategories. Furthermore, we study the category of modules filtered by standard modules for certain quasi-hereditary algebras and we classify precisely when this category...

We introduce the right (left) Gorenstein subcategory relative to an additive subcategory $𝒞$ of an abelian category $𝒜$, and prove that the right Gorenstein subcategory $r𝒢\left(𝒞\right)$ is closed under extensions, kernels of epimorphisms, direct summands and finite direct sums. When $𝒞$ is self-orthogonal, we give a characterization for objects in $r𝒢\left(𝒞\right)$, and prove that any object in $𝒜$ with finite $r𝒢\left(𝒞\right)$-projective dimension is isomorphic to a kernel (or a cokernel) of a morphism from an object in $𝒜$ with finite $𝒞$-projective dimension...