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1.
Every module over an Iwanaga–Gorenstein ring has a Gorenstein flat cover [13] (however, only a few nontrivial examples are
known). Integral group rings over polycyclic-by-finite groups are Iwanaga–Gorenstein [10] and so their modules have such covers.
In particular, modules over integral group rings of finite groups have these covers. In this article we initiate a study of
these covers over these group rings. To do so we study the so-called Gorenstein cotorsion modules, i.e. the modules that split
under Gorenstein flat modules. When the ring is ℤ, these are just the usual cotorsion modules. Harrison [16] gave a complete
characterization of torsion free cotorsion ℤ-modules. We show that with appropriate modifications Harrison's results carry
over to integral group rings ℤG when G is finite. So we classify the Gorenstein cotorsion modules which are also Gorenstein flat over these ℤG. Using these results we classify modules that can be the kernels of Gorenstein flat covers of integral group rings of finite
groups. In so doing we necessarily give examples of such covers. We use the tools we develop to associate an integer invariant
n with every finite group G and prime p. We show 1≤n≤|G : P| where P is a Sylow p-subgroup of G and gives some indication of the significance of this invariant. We also use the results of the paper to describe the co-Galois
groups associated to the Gorenstein flat cover of a ℤG-module.
Presented by A. Verschoren
Mathematics Subject Classifications (2000) 20C05, 16E65. 相似文献
2.
3.
《代数通讯》2013,41(11):4415-4432
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. 相似文献
4.
Driss Bennis 《代数通讯》2013,41(3):855-868
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. 相似文献
5.
Let R be a graded ring and n ? 1 be an integer. We introduce and study the notions of Gorenstein n-FP-gr-injective and Gorenstein n-gr-flat modules by using the notion of special finitely presented graded modules. On n-gr-coherent rings, we investigate the relationships between Gorenstein n-FP-gr-injective and Gorenstein n-gr-flat modules. Among other results, we prove that any graded module in R-gr (or gr-R) admits a Gorenstein n-FP-gr-injective (or Gorenstein n-gr-flat) cover and preenvelope, respectively.
相似文献6.
Guoqiang Zhao 《代数通讯》2013,41(8):3044-3062
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. 相似文献
7.
We give sufficient conditions on a class of R‐modules $\mathcal {C}We give sufficient conditions on a class of R‐modules $\mathcal {C}$ in order for the class of complexes of $\mathcal {C}$‐modules, $dw \mathcal {C}$, to be covering in the category of complexes of R‐modules. More precisely, we prove that if $\mathcal {C}$ is precovering in R ? Mod and if $\mathcal {C}$ is closed under direct limits, direct products, and extensions, then the class $dw \mathcal {C}$ is covering in Ch(R). Our first application concerns the class of Gorenstein flat modules. We show that when the ring R is two sided noetherian, a complex C is Gorenstein flat if and only if each module Cn is Gorenstein flat. If moreover every direct product of Gorenstein flat modules is a Gorenstein flat module, then the class of Gorenstein flat complexes is covering. We consider Gorenstein projective complexes as well. We prove that if R is a commutative noetherian ring of finite Krull dimension, then the class of Gorenstein projective complexes coincides with that of complexes of Gorenstein projective modules. We also show that if R is commutative noetherian with a dualizing complex then every right bounded complex has a Gorenstein projective precover. 相似文献
8.
《Journal of Pure and Applied Algebra》2023,227(1):107127
We extend the notion of virtually Gorenstein rings to the setting of arbitrary rings, and prove that all rings R of finite Gorenstein weak global dimension are virtually Gorenstein such that all Gorenstein projective R-modules are Gorenstein flat. For such a ring R, we introduce the notion of relative homology functors of complexes with respect to Gorenstein projective (resp., flat) modules, and establish a balanced and a vanishing result for the homology functor. 相似文献
9.
《Quaestiones Mathematicae》2013,36(1):35-44
We prove the existence of Gorenstein injective envelopes and covers over n-perfect rings for some classes of modules associated with a dualizing bimodule. 相似文献
10.
Zenghui Gao 《代数通讯》2013,41(8):3035-3044
This article continues to investigate a particular case of Gorenstein FP-injective modules, called strongly Gorenstein FP-injective modules. Some examples are given to show that strongly Gorenstein FP-injective modules lie strictly between FP-injective modules and Gorenstein FP-injective modules. Various results are developed, many extending known results in [1]. We also characterize FC rings in terms of strongly Gorenstein FP-injective, projective, and flat modules. 相似文献
11.
We prove that the class of Gorenstein projective modules is special precovering over any left GF-closed ring such that every Gorenstein projective module is Gorenstein flat and every Gorenstein flat module has finite Gorenstein projective dimension. This class of rings includes (strictly) Gorenstein rings, commutative noetherian rings of finite Krull dimension, as well as right coherent and left n-perfect rings. In Sect. 4 we give examples of left GF-closed rings that have the desired properties (every Gorenstein projective module is Gorenstein flat and every Gorenstein flat has finite Gorenstein projective dimension) and that are not right coherent. 相似文献
12.
13.
Alina Iacob 《代数通讯》2017,45(5):2238-2244
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. 相似文献
14.
《代数通讯》2013,41(4):1453-1470
Abstract In this paper, we show that if R is a local Cohen–Macaulay ring admitting a dualizing module Ω, then Ω-Gorenstein projective and flat covers and Ω-Gorenstein injective envelopes exist for certain modules. These results generalize the well known results for local Gorenstein rings. 相似文献
15.
A right R-module M is called glat if any homomorphism from any finitely presented right R-module to M factors through a finitely presented Gorenstein projective right R-module. The concept of glat modules may be viewed as another Gorenstein analogue of flat modules. We first prove that the class of glat right R-modules is closed under direct sums, direct limits, pure quotients and pure submodules for arbitrary ring R. Then we obtain that a right R-module M is glat if and only if M is a direct limit of finitely presented Gorenstein projective right R-modules. In addition, we explore the relationships between glat modules and Gorenstein flat (Gorenstein projective) modules. Finally we investigate the existence of preenvelopes and precovers by glat and finitely presented Gorenstein projective modules. 相似文献
16.
A complex (C, δ) is called strongly Gorenstein flat if C is exact and Ker δ n is Gorenstein flat in R-Mod for all n ∈ ?. Let 𝒮𝒢 stand for the class of strongly Gorenstein flat complexes. We show that a complex C of left R-modules over a right coherent ring R is in the right orthogonal class of 𝒮𝒢 if and only if C n is Gorenstein cotorsion in R-Mod for all n ∈ ? and Hom.(G, C) is exact for any strongly Gorenstein flat complex G. Furthermore, a bounded below complex C over a right coherent ring R is in the right orthogonal class of 𝒮𝒢 if and only if C n is Gorenstein cotorsion in R-Mod for all n ∈ ?. Finally, strongly Gorenstein flat covers and 𝒮𝒢⊥-envelopes of complexes are considered. For a right coherent ring R, we show that every bounded below complex has a 𝒮𝒢⊥-envelope. 相似文献
17.
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. 相似文献
18.
Let R be a left coherent ring. We first prove that a right R-module M is strongly copure flat if and only if Ext i (M, C) = 0 for all flat cotorsion right R-modules C and i ≥ 1. Then we define and investigate copure flat dimensions of left coherent rings. Finally, we give some new characterizations of n-FC rings. 相似文献
19.
Driss Bennis 《代数通讯》2013,41(10):3837-3850
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. 相似文献
20.
Chaoling Huang 《印度理论与应用数学杂志》2011,42(5):261-277
In this paper, we consider the rings over which the class of finitely generated strongly Gorenstein projective modules is
closed under extensions (called fs-closed rings). We give a characterization about the Grothendieck groups of the category
of the finitely generated strongly Gorenstein projective R-modules and the category of the finitely generated R-modules with finite strongly Gorenstein projective dimensions for any left Noetherian fs-closed ring R. 相似文献