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
The notion of absolutely clean N-complexes is studied.We show that an N-complex X is absolutely clean if and only if X is Nexact and Z,(X)is an absolutely clean module for each n e Z and i=1,2,..,N.In particular,we pr...The notion of absolutely clean N-complexes is studied.We show that an N-complex X is absolutely clean if and only if X is Nexact and Z,(X)is an absolutely clean module for each n e Z and i=1,2,..,N.In particular,we prove that a bounded above N-complex X is absolutely clean if and only if X,is an absolutely clean module for each n e Z.We also show that under certain hypotheses,an Ncomplex X is Gorenstein AC-injective if and only if Z;(X)is a Gorenstein AC-injective module for each n e Z and t=1,2,.,N.展开更多
In this paper we study Gorenstein(semiheredity)heredity,finite presentedness and F P-injectivity of modules over a formal triangular matrix ring.We provide necessary and sufficient conditions for such rings to be Gore...In this paper we study Gorenstein(semiheredity)heredity,finite presentedness and F P-injectivity of modules over a formal triangular matrix ring.We provide necessary and sufficient conditions for such rings to be Gorenstein(semihereditary)hereditary and investigate when a triangular matrix ring is an n-FC ring.展开更多
Let ∧_(0,0)=(A_BMAANB_B)be a Morita ring,where the bimodule homomorphisms φ and ψ are zero.We study the finite presentedness,locally coherence,pure projectivity,pure injectivity,and FP-injectivity of modules over A...Let ∧_(0,0)=(A_BMAANB_B)be a Morita ring,where the bimodule homomorphisms φ and ψ are zero.We study the finite presentedness,locally coherence,pure projectivity,pure injectivity,and FP-injectivity of modules over A_(0,0).Some applications are then given.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
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 an...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.展开更多
Let R and S be a left coherent ring and a right coherent ring respectively,RωS be a faithfully balanced self-orthogonal bimodule.We give a sufficient condition to show that l.FP-idR（ω） ∞ implies G-dimω（M） ∞,w...Let R and S be a left coherent ring and a right coherent ring respectively,RωS be a faithfully balanced self-orthogonal bimodule.We give a sufficient condition to show that l.FP-idR（ω） ∞ implies G-dimω（M） ∞,where M ∈ modR.This result generalizes the result by Huang and Tang about the relationship between the FP-injective dimension and the generalized Gorenstein dimension in 2001.In addition,we get that the left orthogonal dimension is equal to the generalized Gorenstein dimension when G-dimω（M） is finite.展开更多
Given a cotorsion pair （X, Y） in an abelian category A, we define cotorsion pairs （XN,dgYN） and （dgXN, YN） in the category CN（A） of N-complexes on A. We prove that if the cotorsion pair （X, Y） is complete an...Given a cotorsion pair （X, Y） in an abelian category A, we define cotorsion pairs （XN,dgYN） and （dgXN, YN） in the category CN（A） of N-complexes on A. We prove that if the cotorsion pair （X, Y） is complete and hereditary in a bicomplete abelian category, then both of the induced cotorsion pairs are complete, compatible and hereditary. We also create complete cotorsion pairs （dwXN, （dwXN）⊥）, （eXXN, （exXN）⊥） and （⊥（dwYN）, dwYN）, （X（exYN）, exYN） in a termwise manner by starting with a cotorsion pair （X,Y） that is cogenerated by a set. As applications of these results, we obtain more abelian model structures from the cotorsion pairs.展开更多
Let R be an arbitrary associated ring.For an integer N≥2 and a self-orthogonal subcategory W of R-modules,we study the notion of Cartan-Eilenberg W N-complexes.We show that an N-complex X is Cartan-Eilenberg W if and...Let R be an arbitrary associated ring.For an integer N≥2 and a self-orthogonal subcategory W of R-modules,we study the notion of Cartan-Eilenberg W N-complexes.We show that an N-complex X is Cartan-Eilenberg W if and only if X≌X′■X″in which X′is a W iV-complex and X″is a graded/7-module with X″n∈W for all n∈Z.As applications of the result,we obtain some characterizations of Cartan-Eilenberg projective and injective N-complexes,establish Cartan and Eilenberg balance of N-complexes,and give some examples for some fixed integers N to illustrate our main results.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 (12061061)Fundamental Research Funds for the Central Universities (31920230173)+1 种基金Longyuan Young Talents of Gansu ProvinceYoung Talents Team Project of Gansu Province (2025QNTD49)。
文摘The notion of absolutely clean N-complexes is studied.We show that an N-complex X is absolutely clean if and only if X is Nexact and Z,(X)is an absolutely clean module for each n e Z and i=1,2,..,N.In particular,we prove that a bounded above N-complex X is absolutely clean if and only if X,is an absolutely clean module for each n e Z.We also show that under certain hypotheses,an Ncomplex X is Gorenstein AC-injective if and only if Z;(X)is a Gorenstein AC-injective module for each n e Z and t=1,2,.,N.
基金Supported by the National Natural Science Foundation of China(Grant No.11671126)。
文摘In this paper we study Gorenstein(semiheredity)heredity,finite presentedness and F P-injectivity of modules over a formal triangular matrix ring.We provide necessary and sufficient conditions for such rings to be Gorenstein(semihereditary)hereditary and investigate when a triangular matrix ring is an n-FC ring.
基金supported in part by the National Natural Science Foundation of China(Grant Nos.11671126,12071120).
文摘Let ∧_(0,0)=(A_BMAANB_B)be a Morita ring,where the bimodule homomorphisms φ and ψ are zero.We study the finite presentedness,locally coherence,pure projectivity,pure injectivity,and FP-injectivity of modules over A_(0,0).Some applications are then given.
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.
基金Supported by the National Natural Science Foundation of China(Grant Nos.1120137711261050)+1 种基金China Postdoctoral Science Foundation(Grant No.2013M541509)Program of Science and Technique of Gansu Province(Grant No.1208RJZA145)
文摘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.
基金Supported by the Ph. D. Program Foundation of Ministry of Education of China (Grant No.200803570003)
文摘Let R and S be a left coherent ring and a right coherent ring respectively,RωS be a faithfully balanced self-orthogonal bimodule.We give a sufficient condition to show that l.FP-idR（ω） ∞ implies G-dimω（M） ∞,where M ∈ modR.This result generalizes the result by Huang and Tang about the relationship between the FP-injective dimension and the generalized Gorenstein dimension in 2001.In addition,we get that the left orthogonal dimension is equal to the generalized Gorenstein dimension when G-dimω（M） is finite.
基金This research was partially supported by the National Natural Science Foundation of China （11361051, 11761060）, Program for New Century Excellent Talents in University （NCET-13-0957）, and Improvement of Young Teachers＇ Scientific Research Ability （NWNU-LKQN-16-5）.
文摘Given a cotorsion pair （X, Y） in an abelian category A, we define cotorsion pairs （XN,dgYN） and （dgXN, YN） in the category CN（A） of N-complexes on A. We prove that if the cotorsion pair （X, Y） is complete and hereditary in a bicomplete abelian category, then both of the induced cotorsion pairs are complete, compatible and hereditary. We also create complete cotorsion pairs （dwXN, （dwXN）⊥）, （eXXN, （exXN）⊥） and （⊥（dwYN）, dwYN）, （X（exYN）, exYN） in a termwise manner by starting with a cotorsion pair （X,Y） that is cogenerated by a set. As applications of these results, we obtain more abelian model structures from the cotorsion pairs.
基金This work was supported by the Fundamental Research Funds for the Central Universities(No.31920190054)the National Natural Science Foundation of China(Grant No.11971388)XBMUYJRC(No.201406).
文摘Let R be an arbitrary associated ring.For an integer N≥2 and a self-orthogonal subcategory W of R-modules,we study the notion of Cartan-Eilenberg W N-complexes.We show that an N-complex X is Cartan-Eilenberg W if and only if X≌X′■X″in which X′is a W iV-complex and X″is a graded/7-module with X″n∈W for all n∈Z.As applications of the result,we obtain some characterizations of Cartan-Eilenberg projective and injective N-complexes,establish Cartan and Eilenberg balance of N-complexes,and give some examples for some fixed integers N to illustrate our main results.
