Related papers: Gorenstein injective precovers, covers, and envelo…

Let R be a commutative noetherian local ring and consider the set of isomorphism classes of indecomposable totally reflexive R-modules. We prove that if this set is finite, then either it has exactly one element, represented by the rank 1…

We prove that for a left and right Noetherian ring $R$, $_RR$ satisfies the Auslander condition if and only if so does every flat left $R$-module, if and only if the injective dimension of the $i$th term in a minimal flat resolution of any…

We give characterizations of Gorenstein projective, Gorenstein flat and Gorenstein injective modules over the group algebra for large families of infinite groups and show that every weak Gorenstein projective, weak Gorenstein flat and weak…

Let $R$ be any ring with identity and Ch($R$) the category of chain complexes of (left) $R$-modules. We show that the Gorenstein AC-projective chain complexes are the cofibrant objects of an abelian model structure on Ch($R$). The model…

In this paper, we examine the relation between certain subclasses of the classes of Gorenstein projective, Gorenstein flat and Gorenstein injective modules over a group algebra, which consist of the cofibrant, cofibrant-flat and fibrant…

Let R be a graded ring and n > 1 an integer. In this paper, We introduce the notions of Ding n-gr-injective and Ding n-gr-flat modules by using of special finitely presented graded modules . Then, some properties of Ding n-gr-injective and…

Let $R$ by a right coherent ring and $R$-Mod denote the category of left $R$-modules. We show that there is an abelian model structure on $R$-Mod whose cofibrant objects are precisely the Gorenstein flat modules. Employing a new method for…

For any ring $R$ and any positive integer $n$, we prove that a left $R$-module is a Gorenstein $n$-syzygy if and only if it is an $n$-syzygy. Over a left and right Noetherian ring, we introduce the notion of the Gorenstein transpose of…

We prove that for all noetherian rings, the level of any homologically bounded complex $M$ with respect to the collection of projective or injective modules is bounded above by the projective dimension of $\bigoplus H(M)$ plus one or the…

The notion of an $\mathcal{L}$ complex (for a given class of $R$-modules $\mathcal{L}$) was introduced by Gillespie: a complex $C$ is called $\mathcal{L}$ complex if $C$ is exact and $\Z_{i}(C)$ is in $\mathcal{L}$ for all $i\in…

A general principle suggests that "anything flat is a directed colimit of countably presentable flats". In this paper, we consider resolutions and coresolutions of modules over a countably coherent ring $R$ (e.g., any coherent ring or any…

For a dualizing module $D$ over a commutative Noetherian ring $R$ with identity, it is known that its Auslander class $\mathscr{A}_D\left(R\right)$ (respectively, Bass class $\mathscr{B}_D\left(R\right)$) is characterized as those…

We introduce the concepts of generalized compatible and cocompatible bimodules in order to characterize Gorenstein projective, injective and flat modules over trivial ring extensions. Let $R\ltimes M$ be a trivial extension of a ring $R$ by…

Let $T_R(M)$ be a tensor ring, where $R$ is a ring and $M$ is an $N$-nilpotent $R$-bimodule. Under certain conditions, we characterize projectively coresolved Gorenstein flat modules over $T_R(M)$, showing that a $T_R(M)$ module $(X,u)$ is…

We prove that for a Frobenius extension, if a module over the extension ring is Gorenstein projective, then its underlying module over the the base ring is Gorenstein projective; the converse holds if the Frobenius extension is either…

We show that the cosupport of a commutative noetherian ring is precisely the set of primes appearing in a minimal pure-injective resolution of the ring. As an application of this, we prove that every countable commutative noetherian ring…

Let R be a ring, X a class of R-modules and n>1 an integer. In this paper, via special finitely presented modules, we introduce the concepts of Gorenstein n-X-injective and n-X-flat modules. And aside, we obtain some equivalent properties…

In this paper, we study a particular case of Gorenstein projective, injective, and flat modules, which we call, respectively, strongly Gorenstein projective, injective, and flat modules. These last three modules give us a new…

In this paper, we introduce and study the projectively coresolved Gorenstein flat dimension of a group $G$ over a commutative ring $R$ and we prove that this dimension enjoys all the properties of the cohomological and the Gorenstein…

It is proved that a module M over a commutative noetherian ring R is injective if Ext^i((R/p)_p,M)=0 holds for every i\ge 1 and every prime ideal p in R. This leads to the following characterization of injective modules: If F is faithfully…