Gorenstein Flat and Cotorsion Dimensions of Unbounded Complexes

In this article, we define and study the Gorenstein flat dimension and Gorenstein cotorsion dimension for unbounded complexes over GF-closed rings by constructions of resolutions of unbounded complexes. The behavior of the dimensions under change of rings is investigated.     

Gorenstein Projective,Injective, and Flat Complexes

Abstract

We study the classification of those finite groups G having a non-inner class preserving automorphism. Criteria for these automorphisms to be inner are established. Let G be a nilpotent-by-nilpotent group and S?∈?Sy l ₂(G). If S is abelian, generalized quaternion or S is dihedral, and in this case G is also metabelian, then Out _c (G)?=?1. If S is generalized quaternion, 𝒵(S)???𝒵(G) and S ₄ is not a homomorphic image of G, then Out _c (G)?=?1. As a consequence, it follows that the normalizer problem of group rings has a positive answer for these groups.     

On Ding Projective Complexes

In the paper, Ding projective modules and Ding projective complexes are considered. In particular, it is proven that Ding projective complexes are precisely the complexes X for which each X_m is a Ding projective R-module for all m ∈ Z.     

Gorenstein Injective and Gorenstein Flat Dimensions Under Base Change

Abstract

The purpose of this paper is to investigate some connections between Gorenstein flat and Gorenstein injective dimensions of complexes over different rings.     

Gorenstein Dimensions over Ring Homomorphisms

In this paper, we study injective dimension and Gorenstein injective dimension over local ring homomorphisms. Some well-known results are generalized. For example, the Bass formula for Gorenstein injective dimension of complexes is extended. As applications, some characterizations of Gorenstein rings are obtained.     

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.     

模的强Gorenstein平坦维数

In the Gorenstein homological theory, Gorenstein projective and Gorenstein injective dimensions play an important and fundamental role. In this paper, we aim at studying the closely related strongly Gorenstein flat and Gorenstein FP-injective dimensions, and show that some characterizations similar to Gorenstein homological dimensions hold for these two dimensions.     

Gorenstein投射、内射和平坦复形

证明了在任意结合环R上,复形C是Gorenstein投射复形当且仅当每个层次的模C~m是Gorenstein投射模,由此给出了复形Gorenstein投射维数的性质刻画.并证明了对于正合复形C,若对于任意投射模Q,函子Hom(-,Q)作用复形C后仍然得到正合复形,则C是Gorenstein投射复形当且仅当对于所有的m∈Z,有Ker(δ_C~m)是Gorenstein投射模.类似地,本文也讨论了关于Gorenstein内射和Gorenstein平坦复形的相应结果.     

强 Gorenstein 平坦维数

This article is concerned with the strongly Gorenstein flat dimensions of modules and rings.We show this dimension has nice properties when the ring is coherent,and extend the well-known Hilbert＇s syzygy theorem to the strongly Gorenstein flat dimensions of rings.Also,we investigate the strongly Gorenstein flat dimensions of direct products of rings and（almost）excellent extensions of rings.     

n-Strongly Gorenstein Projective,Injective and Flat Modules

In this article, we study the relation between m-strongly Gorenstein projective (resp., injective) modules and n-strongly Gorenstein projective (resp., injective) modules whenever m ≠ n, and the homological behavior of n-strongly Gorenstein projective (resp., injective) modules. We introduce the notion of n-strongly Gorenstein flat modules. Then we study the homological behavior of n-strongly Gorenstein flat modules, and the relation between these modules and n-strongly Gorenstein projective (resp., injective) modules.     

On Gorenstein FP-Injective and Gorenstein Flat Complexes

In this article, the concept of Gorenstein FP-injective modules and some related known results are generalized to Gorenstein FP-injective complexes. Moreover, some new characterizations of Gorenstein flat complexes are given. It is also proved that every complex has a Gorenstein flat preenvelope over coherent rings with finite self-FP-injective dimension.     

群的Gorenstein同调维数

设$k$是一个弱维数有限的交换环, $G$是一个群. 本文讨论了群$G$具有有限的Gorenstein同调维数的标准.证明了群$G$的Gorenstein同调维数的有限性与群环$kG$的Gorenstein弱维数的有限性是一致的.进一步,我们给出了Serre定理的一个Gorenstein类比.推广了整环上$G$的Gorenstein同调维数的一些已知结果.     

Gorenstein projective dimensions of complexes

We show that over a right coherent left perfect ring R, a complex C of left R-modules is Gorenstein projective if and only if C ^m is Gorenstein projective in R-Mod for all m ∈ ℤ. Basing on this we show that if R is a right coherent left perfect ring then Gpd(C) = sup{Gpd(C ^m )|m ∈ ℤ} where Gpd(−) denotes Gorenstein projective dimension.     

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.