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 paper,we introduce the notions of_((m,n))-coherent rings and FP_((m,n))-projective modules for nonnegative integers m,n.We prove that(FP_((m,n))-Proj,(FPn-id)_(≤m))is a complete cotorsion pair for any m,n≥0 ...In this paper,we introduce the notions of_((m,n))-coherent rings and FP_((m,n))-projective modules for nonnegative integers m,n.We prove that(FP_((m,n))-Proj,(FPn-id)_(≤m))is a complete cotorsion pair for any m,n≥0 and it is hereditary if and only if the ring R is a left n-coherent ring for all m≥0 and n≥1.Moreover,we study the existence of FP_((m,n))-Proj covers and envelopes and obtain that if FP_((m,n))-Proj is closed under pure quotients,then FP_((m,n))-Proj is covering for any n≥2.As applications,we obtain that every R-module has an epic FP_((m,n))-Proj-envelope if and only if the left FP_((m,n))-global dimension of R is at most 1 and FP_((m,n))-Proj is closed under direct products.展开更多
Let R be a ring, * be an involutory function of the set of all finite matrices over R. In this paper, necessary and sufficient conditions are given for a matrix to have a (1,3)-inverse, (1,4)-inverse, or Moore-P enros...Let R be a ring, * be an involutory function of the set of all finite matrices over R. In this paper, necessary and sufficient conditions are given for a matrix to have a (1,3)-inverse, (1,4)-inverse, or Moore-P enrose inverse, relative to *. Some results about generalized inverses of matrices over division rings are generalized and improved.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(No.12471036),the project of Young and Middle-Aged Talents of Hubei Province(No.Q20234405),and the Scientific Research Fund of Hunan Provincial Education Department(No.24A0221)。
文摘In this paper,we introduce the notions of_((m,n))-coherent rings and FP_((m,n))-projective modules for nonnegative integers m,n.We prove that(FP_((m,n))-Proj,(FPn-id)_(≤m))is a complete cotorsion pair for any m,n≥0 and it is hereditary if and only if the ring R is a left n-coherent ring for all m≥0 and n≥1.Moreover,we study the existence of FP_((m,n))-Proj covers and envelopes and obtain that if FP_((m,n))-Proj is closed under pure quotients,then FP_((m,n))-Proj is covering for any n≥2.As applications,we obtain that every R-module has an epic FP_((m,n))-Proj-envelope if and only if the left FP_((m,n))-global dimension of R is at most 1 and FP_((m,n))-Proj is closed under direct products.
文摘Let R be a ring, * be an involutory function of the set of all finite matrices over R. In this paper, necessary and sufficient conditions are given for a matrix to have a (1,3)-inverse, (1,4)-inverse, or Moore-P enrose inverse, relative to *. Some results about generalized inverses of matrices over division rings are generalized and improved.
