On (Strongly) Gorenstein Von Neumann Regular Rings

In this article, we introduce and study the rings over which every module is (strongly) Gorenstein flat, which we call them (strongly) Gorenstein Von Neumann regular rings.     

First,Second, and Third Change of Rings Theorems for Gorenstein Homological dimensions

In this article, we investigate the change of rings theorems for the Gorenstein dimensions over arbitrary rings. Namely, by the use of the notion of strongly Gorenstein modules, we extend the well-known first, second, and third change of rings theorems for the classical projective and injective dimensions to the Gorenstein projective and injective dimensions, respectively. Each of the results established in this article for the Gorenstein projective dimension is a generalization of a G-dimension of a finitely generated module M over a noetherian ring R.     

The existence of finitely generated modules of finite Gorenstein injective dimension

In this note, we study commutative Noetherian local rings having finitely generated modules of finite Gorenstein injective dimension. In particular, we consider whether such rings are Cohen-Macaulay.

     

Cohomology theories based on Gorenstein injective modules

In this paper we study relative and Tate cohomology of modules of finite Gorenstein injective dimension. Using these cohomology theories, we present variations of Grothendieck local cohomology modules, namely Gorenstein and Tate local cohomology modules. By applying a sort of Avramov-Martsinkovsky exact sequence, we show that these two variations of local cohomology are tightly connected to the generalized local cohomology modules introduced by J. Herzog. We discuss some properties of these modules and give some results concerning their vanishing and non-vanishing.

     

Gorenstein derived functors

Over any associative ring it is standard to derive using projective resolutions in the first variable, or injective resolutions in the second variable, and doing this, one obtains in both cases. We examine the situation where projective and injective modules are replaced by Gorenstein projective and Gorenstein injective ones, respectively. Furthermore, we derive the tensor product using Gorenstein flat modules.

     

Gorenstein flatness and injectivity over Gorenstein rings

Let R be a Gorenstein ring.We prove that if I is an ideal of R such that R/I is a semi-simple ring,then the Gorenstein flat dimension of R/I as a right R-module and the Gorenstein injective dimension of R/I as a left R-module are identical.In addition,we prove that if R→S is a homomorphism of rings and SE is an injective cogenerator for the category of left S-modules,then the Gorenstein flat dimension of S as a right R-module and the Gorenstein injective dimension of E as a left R-module are identical.We also give some applications of these results.     

Gorenstein Analogue of Auslander's Theorem on the Global Dimension

In this paper, we prove that the Gorenstein analogue of the well-known Auslander's theorem on the global dimension holds true. Namely, we prove that the Gorenstein global dimension of a commutative ring R is equal to the supremum of the set of Gorenstein projective dimensions of all cyclic R-modules.     

Gorenstein Injective and Gorenstein Flat Resolution of Modules Over Gorenstein Rings

Abstract

Let R be a commutative Noetherian ring. There are several characterizations of Gorenstein rings in terms of classical homological dimensions of their modules. In this paper, we use Gorenstein dimensions (Gorenstein injective and Gorenstein flat dimension) to describe Gorenstein rings. Moreover a characterization of Gorenstein injective (resp. Gorenstein flat) modules over Gorenstein rings is given in terms of their Gorenstein flat (resp. Gorenstein injective) resolutions.     

Gorenstein injective modules and local cohomology

In this paper we assume that is a Gorenstein Noetherian ring. We show that if is also a local ring with Krull dimension that is less than or equal to 2, then for any nonzero ideal of , is Gorenstein injective. We establish a relation between Gorenstein injective modules and local cohomology. In fact, we will show that if is a Gorenstein ring, then for any -module its local cohomology modules can be calculated by means of a resolution of by Gorenstein injective modules. Also we prove that if is -Gorenstein, is a Gorenstein injective and is a nonzero ideal of , then is Gorenstein injective.

     

Complete cohomology for complexes with finite Gorenstein AC-projective dimension

We study complete cohomology of complexes with finite Gorenstein AC-projective dimension. We show first that the class of complexes admitting a complete level resolution is exactly the class of complexes with finite Gorenstein AC-projective dimension. This lets us give some general techniques for computing complete cohomology of complexes with finite Gorenstein AC-projective dimension. As a consequence, the classical relative cohomology for modules of finite Gorenstein AC-projective dimension is extended. Finally, the relationships between projective dimension and Gorenstein AC-projective dimension for complexes are given.     

Rings Over Which the Class of Gorenstein Flat Modules is Closed Under Extensions

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 dimension.

In this article, we investigate the Gorenstein flat dimension over left GF-closed rings. Namely, we generalize the fact that the class of all Gorenstein flat left modules is projectively resolving over right coherent rings to left GF-closed rings. Also, we generalize the characterization of Gorenstein flat left modules (then of Gorenstein flat dimension of left modules) over right coherent rings to left GF-closed rings. Finally, using direct products of rings, we show how to construct a left GF-closed ring that is neither right coherent nor of finite weak dimension.     

Gorenstein Modules over Zariski Filtered Rings

Abstract

We study Gorenstein injective and projective modules over Zariski filtered rings and obtain relations between the Gorenstein dimensions on the category of filtered modules from the associated category of graded modules over the associated graded ring.     

Gorenstein Flat Covers over GF-Closed Rings

We prove that all modules over a left GF-closed ring have Gorenstein flat covers.     

n- Gorenstein环上的 Gorenstein内射模(英文)

用 Gorenstein内射模刻画了 n-Gorenstein环 .     

Coherent Rings and Gorenstein FP-Injective Modules

In this article, Gorenstein FP-injective modules are introduced and investigated. A left R-module M is called Gorenstein FP-injective if there is an exact sequence … → E ₁ → E ₀ → E ⁰ → E ¹ → … of FP-injective left R-modules with M = ker(E ⁰ → E ¹) such that Hom _R (P, ?) leaves the sequence exact whenever P is a finitely presented left R-module with pd _R (P) < ∞. Some properties of Gorenstein FP-injective modules are obtained. Several well-known classes of rings are characterized in terms of Gorenstein FP-injective modules.     

Gorenstein injective envelopes and covers over two sided noetherian rings

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 consider the connection between the Gorenstein injective modules and the strongly cotorsion modules. We prove that when the ring R is commutative noetherian of finite Krull dimension, the class of Gorenstein injective modules coincides with that of strongly cotorsion modules if and only if the ring R is in fact Gorenstein.     

Gorenstein同调维数和环变换

In this paper,we shall be concerned with what happens of Gorenstein homological dimensions when certain modifications are made to a ring.The five structural operations addressed later are the formation of excellent extensions,localizations,Morita equivalences,polynomial extensions and power series extensions.     

Some Results on Gorenstein Dedekind Domains and Their Factor Rings

A domain is called a Gorenstein Dedekind domain (G-Dedekind for short) if every submodule of a projective module is G-projective (i.e., G-gldim(R) = 1). It is proved in this note that a domain R is a G-Dedekind domain if and only if every ideal of R is Gorenstein-projective (G-projective for short). We also show that nontrivial factor rings of Dedekind domains are QF-rings. We also give an example to show that factor rings of QF-rings are not necessarily QF-rings.