An R-module M is called Gorenstein FP-injective if there is an exact sequence …→E1→E0→E^0→E^1→… of FP-injective R-modules with M=ker（E^0→E^1） and such that Hom（E,-） leaves the sequence exact whenever E is ...An R-module M is called Gorenstein FP-injective if there is an exact sequence …→E1→E0→E^0→E^1→… of FP-injective R-modules with M=ker（E^0→E^1） and such that Hom（E,-） leaves the sequence exact whenever E is an FP-injective R-module.Some properties of Gorenstein FP-injective are obtained.Moreover,it is proved that a ring is left Noetherian if and only if every Gorenstein FP-injective left R-module is Gorenstein injective.Furthermore,it is shown that over an n-FC and perfect ring R,a left R-module M is Gorenstein FP-injective if and only if MFH for some FP-injective left R-module F and some strongly Gorenstein FP-injective R-module H.In view of this,Gorenstein FP-injective precovers and Gorenstein FP-injective preenvelopes are considered.展开更多
In this article, we introduce and study the concept of n-Gorenstein injective （resp., n-Gorenstein flat） modules as a nontrivial generalization of Gorenstein injective （resp., Gorenstein flat） modules. We invest...In this article, we introduce and study the concept of n-Gorenstein injective （resp., n-Gorenstein flat） modules as a nontrivial generalization of Gorenstein injective （resp., Gorenstein flat） modules. We investigate the properties of these modules in various ways. For example, we show that the class of n-Gorenstein injective （resp., n -Gorenstein flat） modules is closed under direct sums and direct products for n ≥ 2. To this end, we first introduce and study the notions of n-injective modules and n-flat modules.展开更多
For a local commutative Gorenstein ring R,Enochs et al.in[Gorenstein projective resolvents,Comm.Algebra 44(2016)3989-4000)defined a functor Extn^(R)(-,-)and showed that this functor can be computed by taking a totally...For a local commutative Gorenstein ring R,Enochs et al.in[Gorenstein projective resolvents,Comm.Algebra 44(2016)3989-4000)defined a functor Extn^(R)(-,-)and showed that this functor can be computed by taking a totally acyclic complex arising from a projective coresolution of the first component or a totally acyclic complex arising from a projective resolution of the second component.In order to define the functor Extn^(R)(-,-)over general rings,we introduce the right Gorenstein projective dimension of an R-module M,RGpd(M),via Gorenstein projective coresolutions,and give some equivalent characterizations for the finiteness of RGpd(M).Then over a general ring R we define a co-Tate homology group Extn^(R)(-,-) for R-modules M and N with RGpd(M)<oo and Gpd(N)<∞,and prove that Extn^(R)(M,N)can be computed by complete projective coresolutions of the first variable or by complete projective resolutions of the second variable.展开更多
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 E Z. Basing on this we show that if R is a ...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 E 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.展开更多
基金The National Natural Science Foundation of China (No.10971024)Specialized Research Fund for the Doctoral Program of Higher Education (No. 200802860024)
文摘An R-module M is called Gorenstein FP-injective if there is an exact sequence …→E1→E0→E^0→E^1→… of FP-injective R-modules with M=ker（E^0→E^1） and such that Hom（E,-） leaves the sequence exact whenever E is an FP-injective R-module.Some properties of Gorenstein FP-injective are obtained.Moreover,it is proved that a ring is left Noetherian if and only if every Gorenstein FP-injective left R-module is Gorenstein injective.Furthermore,it is shown that over an n-FC and perfect ring R,a left R-module M is Gorenstein FP-injective if and only if MFH for some FP-injective left R-module F and some strongly Gorenstein FP-injective R-module H.In view of this,Gorenstein FP-injective precovers and Gorenstein FP-injective preenvelopes are considered.
基金The NSF(11501451)of Chinathe Fundamental Research Funds(31920150038)for the Central Universities and XBMUYJRC(201406)
文摘In this article, we introduce and study the concept of n-Gorenstein injective （resp., n-Gorenstein flat） modules as a nontrivial generalization of Gorenstein injective （resp., Gorenstein flat） modules. We investigate the properties of these modules in various ways. For example, we show that the class of n-Gorenstein injective （resp., n -Gorenstein flat） modules is closed under direct sums and direct products for n ≥ 2. To this end, we first introduce and study the notions of n-injective modules and n-flat modules.
基金Supported by National Natural Science Foundation of China(Grant No.11971388).
文摘For a local commutative Gorenstein ring R,Enochs et al.in[Gorenstein projective resolvents,Comm.Algebra 44(2016)3989-4000)defined a functor Extn^(R)(-,-)and showed that this functor can be computed by taking a totally acyclic complex arising from a projective coresolution of the first component or a totally acyclic complex arising from a projective resolution of the second component.In order to define the functor Extn^(R)(-,-)over general rings,we introduce the right Gorenstein projective dimension of an R-module M,RGpd(M),via Gorenstein projective coresolutions,and give some equivalent characterizations for the finiteness of RGpd(M).Then over a general ring R we define a co-Tate homology group Extn^(R)(-,-) for R-modules M and N with RGpd(M)<oo and Gpd(N)<∞,and prove that Extn^(R)(M,N)can be computed by complete projective coresolutions of the first variable or by complete projective resolutions of the second variable.
基金Supported by National＇ Natural Science Foundation of China （Grant No. 10961021）, TRAPOYT and the Cultivation Fund of the Key Scientific and Technical Innovation Project, Ministry of Education of China
文摘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 E 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.
