Related papers: Gorenstein projective, injective and flat modules …

We show that a differential module is Gorenstein projective if and only if its underlying module is Gorenstein projective. Dually, a differential module is Gorenstein injective if and only if its underlying module is Gorenstein injective.

We characterize Ding modules and complexes over Ding-Chen rings. We show that over a Ding-Chen ring R, the Ding projective (resp. Ding injective, resp. Ding flat) R-modules coincide with the Gorenstein projective (resp. Gorenstein…

In this paper, we prove that the global Gorenstein projective dimension of a ring $R$ is equal to the global Gorenstein injective dimension of $R$, and that the global Gorenstein flat dimension of $R$ is smaller than the common value of the…

Let $T=\left( \begin{array}{cc} R & M 0 & S \end{array} \right) $ be a triangular matrix ring with $R$ and $S$ rings and $_RM_S$ an $R$-$S$-bimodule. We describe Gorenstein projective modules over $T$. In particular, we refine a result of…

A recent result by J. \v{S}aroch and J. \v{S}\v{t}ov\'{\i}\v{c}ek asserts that there is a unique abelian model structure on the category of left $R$-modules, for any associative ring $R$ with identity, whose (trivially) cofibrant and…

This paper generalize the idea of the authors in \cite{Bennis and Mahdou1}. Namely, we define and study a particular case of modules with Gorenstein projective, injective, and flat dimension less or equal than $n\geq 0$, which we call,…

For a $k$-algebra $A$, a quiver $Q$, and an ideal $I$ of $kQ$ generated by monomial relations, let $\Lambda: = A\otimes_k kQ/I$. We introduce the monic representations of $(Q, I)$ over $A$. We give properties of the structural maps of monic…

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…

An A-module M will be said to be semi-Gorenstein-projective provided that Ext^i(M,A) = 0 for all i > 0. All Gorenstein-projective modules are semi-Gorenstein-projective and only few and quite complicated examples of…

We consider a right coherent ring R. We prove that the class of Gorenstein flat complexes is covering in the category of complexes of left R-modules Ch(R). When R is also left n-perfect, we prove that the class of Gorenstein projective…

We prove that the class of Gorenstein injective modules is both enveloping and covering over a two sided noetherian ring such that the character modules of Gorenstein injective modules are Gorenstein flat. In the second part of the paper we…

A left $R$-module $M$ is called two-degree Ding projective if there exists an exact sequence $...\longrightarrow D_{1}\longrightarrow D_{0}\longrightarrow D_{-1}\longrightarrow D_{-2}\longrightarrow...$ of Ding projective left $R$-modules…

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…

We formulate and answer Gorenstein projective, flat, and injective analogues of a classical projectivity question for group rings under some mild additional assumptions. Although the original question, that was proposed by Jang-Hyun Jo in…

Let $A$ be a coherent algebra and $B$ be a finite-dimensional Gorenstein algebra over a field $k$. We describe finitely presented Gorenstein projective $A\otimes_k B$-modules in terms of their underlying onesided modules. Moreover, if the…

Let $(\mathcal{A,B})$ be a complete and hereditary cotorsion pair in the category of left $R$-modules. In this paper, the so-called Gorenstein projective complexes respect to the cotorsion pair $(\mathcal{A}, \mathcal{B})$ are introduced.…

A ring $R$ is called left GF-closed, if the class of all Gorenstein flat left $R$-modules is closed under extensions. The class of left GF-closed rings includes strictly the one of right coherent rings and the one of rings of finite weak…

Let $R$ be a two-sided noetherian ring and $M$ be a nilpotent $R$-bimodule, which is finitely generated on both sides. We study Gorenstein homological properties of the tensor ring $T_R(M)$. Under certain conditions, the ring $R$ is…

Gorenstein homological dimensions are refinements of the classical homological dimensions, and finiteness singles out modules with amenable properties reflecting those of modules over Gorenstein rings. As opposed to their classical…

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…