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
As a proper setting to study Gorenstein projective and injective dimensions of modules via vanishing of Gorenstein Ext-functors, a notion of a generalized Gorenstein ring is introduced, which is a non-trivial generali...As a proper setting to study Gorenstein projective and injective dimensions of modules via vanishing of Gorenstein Ext-functors, a notion of a generalized Gorenstein ring is introduced, which is a non-trivial generalization of Gorenstein rings. Moreover, a new proof for Bennis and Mahdou's equality of global Gorenstein dimension is given.展开更多
The representation of weak Hopf algebras is studied by investigating the Gorenstein dimensions of weak Hopf algebras and weak Hopf-Galois extensions.Let H be a weak Hopf algebra with a bijective antipode,A a weak righ...The representation of weak Hopf algebras is studied by investigating the Gorenstein dimensions of weak Hopf algebras and weak Hopf-Galois extensions.Let H be a weak Hopf algebra with a bijective antipode,A a weak right H-comodule algebra and B the H-coinvariant subalgebra of A.First,some properties of Gorenstein projective H-modules in the representation category are studied,and the fact that Gorenstein global dimension of H is the same as the Gorenstein projective dimension of its left unital subalgebra is demonstrated.Secondly,by applying the integral theory of weak Hopf algebras,on the one hand,a sufficient and necessary condition that a projective A-module is a projective B-module is given;on the other hand,the separability of the functor AB-and that of the restriction of scalar function B(-)are described,respectively.Finally,as a mean result,the Gorenstein global dimension of a weak Hopf-Galois extension is investigated under the condition that H is both semisimple and cosemisimple.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Supported by the National Natural Science Foundation of China（11401476） Supported by the Project for Universities of Gansu Province（2015A-019）
文摘As a proper setting to study Gorenstein projective and injective dimensions of modules via vanishing of Gorenstein Ext-functors, a notion of a generalized Gorenstein ring is introduced, which is a non-trivial generalization of Gorenstein rings. Moreover, a new proof for Bennis and Mahdou's equality of global Gorenstein dimension is given.
基金The National Natural Science Foundation of China(No.11601203)the China Postdoctoral Science Foundation(No.2018M642128)Qing Lan Project of Jiangsu Province,the Natural Science Foundation of Jiangsu Province(No.BK20150113).
文摘The representation of weak Hopf algebras is studied by investigating the Gorenstein dimensions of weak Hopf algebras and weak Hopf-Galois extensions.Let H be a weak Hopf algebra with a bijective antipode,A a weak right H-comodule algebra and B the H-coinvariant subalgebra of A.First,some properties of Gorenstein projective H-modules in the representation category are studied,and the fact that Gorenstein global dimension of H is the same as the Gorenstein projective dimension of its left unital subalgebra is demonstrated.Secondly,by applying the integral theory of weak Hopf algebras,on the one hand,a sufficient and necessary condition that a projective A-module is a projective B-module is given;on the other hand,the separability of the functor AB-and that of the restriction of scalar function B(-)are described,respectively.Finally,as a mean result,the Gorenstein global dimension of a weak Hopf-Galois extension is investigated under the condition that H is both semisimple and cosemisimple.
