Let R be a commutative ring and (S, ≤) a strictly totally ordered monoid which satisfies the condition that 0 ≤ s for every s ∈ S. In this paper we show that if RM is a PS-module, then the module [[MS≤]] of genera...Let R be a commutative ring and (S, ≤) a strictly totally ordered monoid which satisfies the condition that 0 ≤ s for every s ∈ S. In this paper we show that if RM is a PS-module, then the module [[MS≤]] of generalized power series over M is a PS [[RS,≤]]-module.展开更多
Let R be a ring, and let （F, C） be a cotorsion theory. In this article, the notion of F-perfect rings is introduced as a nontrial generalization of perfect rings and A-perfect rings. A ring R is said to be right dr-...Let R be a ring, and let （F, C） be a cotorsion theory. In this article, the notion of F-perfect rings is introduced as a nontrial generalization of perfect rings and A-perfect rings. A ring R is said to be right dr-perfect if F is projective relative to R for any F ∈ F. We give some characterizations of F-perfect rings. For example, we show that a ring R is right F-perfect if and only if F-covers of finitely generated modules are projective. Moreover, we define F-perfect modules and investigate some properties of them.展开更多
A left ideal I of a ring R is small in case for every proper left ideal K of R, K ＋ I ≠R. A ring R is called left PS-coherent if every principally small left ideal Ra is finitely presented. We develop, in this paper...A left ideal I of a ring R is small in case for every proper left ideal K of R, K ＋ I ≠R. A ring R is called left PS-coherent if every principally small left ideal Ra is finitely presented. We develop, in this paper, PS-coherent rings as a generalization of P-coherent rings and J-coherent rings. To characterize PS-coherent rings, we first introduce PS-injective and PS-flat modules, and discuss the relation between them over some spacial rings. Some properties of left PS-coherent rings are also studied.展开更多
In this paper, we discuss FI-extending property of rings and modules. The main results are the following： a characterization of von Neumann regular rings which are two-sided FI-extending is given; sufficient conditio...In this paper, we discuss FI-extending property of rings and modules. The main results are the following： a characterization of von Neumann regular rings which are two-sided FI-extending is given; sufficient conditions for direct summands of FI-extending modules to be FI-extending are obtained; and at last, a necessary and sufficient condition for nonsingular modules over nonsingular rings to be FI-extending is given.展开更多
We introduce and investigate the concept of s-injective modules and strongly s-injective modules. New characterizations of SI-rings, GV-rings and pseudo-Frobenius rings are given in terms of s-injectivity of their mod...We introduce and investigate the concept of s-injective modules and strongly s-injective modules. New characterizations of SI-rings, GV-rings and pseudo-Frobenius rings are given in terms of s-injectivity of their modules.展开更多
Let T be a formal triangular matrix ring.We prove that,if for each 1≤j<i≤n,U_(ij) is flat on both sides,then a left T-module■is Ding projective if and only if M1 is a Ding projective left A1-module and for each ...Let T be a formal triangular matrix ring.We prove that,if for each 1≤j<i≤n,U_(ij) is flat on both sides,then a left T-module■is Ding projective if and only if M1 is a Ding projective left A1-module and for each 1≤k≤n-1 the mappingФk+1,k:U_(k+1),k■AkM_(k)→M_(k)+1 is injective with cokernel Ding projective over Ak+1.As a consequence,we describe Ding projective dimension of a left T-module.展开更多
Let U be a (B, A)-bimodule, A and B be rings, and be a formal triangular matrix ring. In this paper, we characterize the structure of relative Ding projective modules over T under some conditions. Furthermore, using t...Let U be a (B, A)-bimodule, A and B be rings, and be a formal triangular matrix ring. In this paper, we characterize the structure of relative Ding projective modules over T under some conditions. Furthermore, using the left global relative Ding projective dimensions of A and B, we estimate the relative Ding projective dimension of a left T-module.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
Some homological properties of R-modules were investigated by matrices over aring R. Given two cardinal numbers α, β and an α x β row-finite matrix A, it was proved thatExt_R^1(R^((α))/R^((β))A, M) = 0 if and on...Some homological properties of R-modules were investigated by matrices over aring R. Given two cardinal numbers α, β and an α x β row-finite matrix A, it was proved thatExt_R^1(R^((α))/R^((β))A, M) = 0 if and only if M_α/r_(M_α)(R^((β))A) ≈ Hom_R(R^((β))A,M) ifand only if r_(M_β)l_(R^((β)))(A) = AM_α. Thus, the notion of (m,n)-injectivity was extended.Moreover, ( α, β) -flatness was characterized via annihilators of matrices, factorizations ofhomomorphisms as well as homological groups so that (m, n)-flat modules, f-projective modules andn-projective modules were consolidated under the notion of (α, β)-flat modules. Furthermore, acharacterization of left R-ML modules and some equivalent conditions for R^((β)) to be left R-MLwere presented. Consequently, the notions of coherent rings, (m, n)-coherent rings and π-coherentrings were consolidated under that of (α, β)-coherent rings.展开更多
We first introduce the concepts of absolutely E-pure modules and E-pure split modules. Then, we characterize the IF rings in terms of absolutely E-pure modules. The E-pure split modules are also characterized.
A ring R is called right zip provided that if the annihilator τR（X） of a subset X of R is zero, then τR（Y） = 0 for some finite subset Y C X. Such rings have been studied in literature. For a right R-module M, we...A ring R is called right zip provided that if the annihilator τR（X） of a subset X of R is zero, then τR（Y） = 0 for some finite subset Y C X. Such rings have been studied in literature. For a right R-module M, we introduce the notion of a zip module, which is a generalization of the right zip ring. A number of properties of this sort of modules are established, and the equivalent conditions of the right zip ring R are given. Moreover, the zip properties of matrices and polynomials over a module M are studied.展开更多
Let R be a Noetherian ring. The projectivity and injectivity of modules over R are discussed. The concept of modules is introduced and the descriptions for co-*-modules over R are given. At last, cotilting modules ove...Let R be a Noetherian ring. The projectivity and injectivity of modules over R are discussed. The concept of modules is introduced and the descriptions for co-*-modules over R are given. At last, cotilting modules over R are characterized by means of co-*-modules.展开更多
The aim of this paper is to investigate higher level orderings on modulesover commutative rings. On the basis of the theory of higher level orderings on fields andcommutative rings, some results involving existence of...The aim of this paper is to investigate higher level orderings on modulesover commutative rings. On the basis of the theory of higher level orderings on fields andcommutative rings, some results involving existence of higher level orderings are generalized to thecategory of modules over commutative rings. Moreover, a strict intersection theorem for higherlevel orderings on modules is established.展开更多
Let A and B be rings and U a(B,A)-bimodule.If BU is flat and UA is finitely generated projective(resp.,BU is finitely generated projective and UA is flat),then the characterizations of level modules and Gorenstein AC-...Let A and B be rings and U a(B,A)-bimodule.If BU is flat and UA is finitely generated projective(resp.,BU is finitely generated projective and UA is flat),then the characterizations of level modules and Gorenstein AC-projective modules(resp.,absolutely clean modules and Gorenstein AC-injective modules)over the formal triangular matrix ring T=(A0 UB)are given.As applications,it is proved that every Gorenstein AC-projective left T-module is projective if and only if each Gorenstein AC-projective left A-module and B-module is projective,and every Gorenstein AC-injective left T-module is injective if and only if each Gorenstein AC-injective left A-module and B-module is injective.Moreover,Gorenstein AC-projective and AC-injective dimensions over the formal triangular matrix ring T are studied.展开更多
Ring epimorphisms often induce silting modules and cosilting modules,termed minimal silting or minimal cosilting.The aim of this paper is twofold.Firstly,we determine the minimal tilting and minimal cotilting modules ...Ring epimorphisms often induce silting modules and cosilting modules,termed minimal silting or minimal cosilting.The aim of this paper is twofold.Firstly,we determine the minimal tilting and minimal cotilting modules over a tame hereditary algebra.In particular,we show that a large cotilting module is minimal if and only if it has an adic module as a direct summand.Secondly,we discuss the behavior of minimality under ring extensions.We show that minimal cosilting modules over a commutative noetherian ring extend to minimal cosilting modules along any flat ring epimorphism.Similar results are obtained for commutative rings of small homological dimensions.展开更多
Let R be a commutative Noetherian ring and p be a prime ideal of R such that the ideal pRp is principal and ht(p)≠0. In this note, the anthors describe the explicit structure of the injective envelope of the R-module...Let R be a commutative Noetherian ring and p be a prime ideal of R such that the ideal pRp is principal and ht(p)≠0. In this note, the anthors describe the explicit structure of the injective envelope of the R-module R/p.展开更多
In this paper, we introduce the concept of almost cotorsion modules. A module is called almost cotorsion if it is subisomorphic to its cotorsion envelope. Some characterizations of almost cotorsion modules are given. ...In this paper, we introduce the concept of almost cotorsion modules. A module is called almost cotorsion if it is subisomorphic to its cotorsion envelope. Some characterizations of almost cotorsion modules are given. It is also proved that every module is a direct summand of an almost cotorsion module. As an application, perfect rings are characterized in terms of almost cotorsion modules.展开更多
基金The NNSF (10171082) of China and the Teaching and Research Award Program for Outstanding Young Teachers in Higher Education Institutions of MOE, P.R.C.
文摘Let R be a commutative ring and (S, ≤) a strictly totally ordered monoid which satisfies the condition that 0 ≤ s for every s ∈ S. In this paper we show that if RM is a PS-module, then the module [[MS≤]] of generalized power series over M is a PS [[RS,≤]]-module.
文摘Let R be a ring, and let （F, C） be a cotorsion theory. In this article, the notion of F-perfect rings is introduced as a nontrial generalization of perfect rings and A-perfect rings. A ring R is said to be right dr-perfect if F is projective relative to R for any F ∈ F. We give some characterizations of F-perfect rings. For example, we show that a ring R is right F-perfect if and only if F-covers of finitely generated modules are projective. Moreover, we define F-perfect modules and investigate some properties of them.
文摘A left ideal I of a ring R is small in case for every proper left ideal K of R, K ＋ I ≠R. A ring R is called left PS-coherent if every principally small left ideal Ra is finitely presented. We develop, in this paper, PS-coherent rings as a generalization of P-coherent rings and J-coherent rings. To characterize PS-coherent rings, we first introduce PS-injective and PS-flat modules, and discuss the relation between them over some spacial rings. Some properties of left PS-coherent rings are also studied.
基金The NNSF(10571026)of Chinathe NSF(2005207)of Jiangsu Provincethe Specialized Research Fund(20060286006)for the Doctoral Program of Higher Education.
文摘In this paper, we discuss FI-extending property of rings and modules. The main results are the following： a characterization of von Neumann regular rings which are two-sided FI-extending is given; sufficient conditions for direct summands of FI-extending modules to be FI-extending are obtained; and at last, a necessary and sufficient condition for nonsingular modules over nonsingular rings to be FI-extending is given.
文摘We introduce and investigate the concept of s-injective modules and strongly s-injective modules. New characterizations of SI-rings, GV-rings and pseudo-Frobenius rings are given in terms of s-injectivity of their modules.
基金Supported by the National Natural Science Foundation of China(Grant No.11861055)。
文摘Let T be a formal triangular matrix ring.We prove that,if for each 1≤j<i≤n,U_(ij) is flat on both sides,then a left T-module■is Ding projective if and only if M1 is a Ding projective left A1-module and for each 1≤k≤n-1 the mappingФk+1,k:U_(k+1),k■AkM_(k)→M_(k)+1 is injective with cokernel Ding projective over Ak+1.As a consequence,we describe Ding projective dimension of a left T-module.
文摘Let U be a (B, A)-bimodule, A and B be rings, and be a formal triangular matrix ring. In this paper, we characterize the structure of relative Ding projective modules over T under some conditions. Furthermore, using the left global relative Ding projective dimensions of A and B, we estimate the relative Ding projective dimension of a left T-module.
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Some homological properties of R-modules were investigated by matrices over aring R. Given two cardinal numbers α, β and an α x β row-finite matrix A, it was proved thatExt_R^1(R^((α))/R^((β))A, M) = 0 if and only if M_α/r_(M_α)(R^((β))A) ≈ Hom_R(R^((β))A,M) ifand only if r_(M_β)l_(R^((β)))(A) = AM_α. Thus, the notion of (m,n)-injectivity was extended.Moreover, ( α, β) -flatness was characterized via annihilators of matrices, factorizations ofhomomorphisms as well as homological groups so that (m, n)-flat modules, f-projective modules andn-projective modules were consolidated under the notion of (α, β)-flat modules. Furthermore, acharacterization of left R-ML modules and some equivalent conditions for R^((β)) to be left R-MLwere presented. Consequently, the notions of coherent rings, (m, n)-coherent rings and π-coherentrings were consolidated under that of (α, β)-coherent rings.
文摘We first introduce the concepts of absolutely E-pure modules and E-pure split modules. Then, we characterize the IF rings in terms of absolutely E-pure modules. The E-pure split modules are also characterized.
基金The NNSF (10571026) of Chinathe Specialized Research Fund (20060286006) for the Doctoral Program of Higher Education.
文摘A ring R is called right zip provided that if the annihilator τR（X） of a subset X of R is zero, then τR（Y） = 0 for some finite subset Y C X. Such rings have been studied in literature. For a right R-module M, we introduce the notion of a zip module, which is a generalization of the right zip ring. A number of properties of this sort of modules are established, and the equivalent conditions of the right zip ring R are given. Moreover, the zip properties of matrices and polynomials over a module M are studied.
文摘Let R be a Noetherian ring. The projectivity and injectivity of modules over R are discussed. The concept of modules is introduced and the descriptions for co-*-modules over R are given. At last, cotilting modules over R are characterized by means of co-*-modules.
基金1)Project supported hy the National Natural Science Foundation of China,Grant No,19661002
文摘The aim of this paper is to investigate higher level orderings on modulesover commutative rings. On the basis of the theory of higher level orderings on fields andcommutative rings, some results involving existence of higher level orderings are generalized to thecategory of modules over commutative rings. Moreover, a strict intersection theorem for higherlevel orderings on modules is established.
基金partly supported by NSF of China(grants 11761047 and 11861043).
文摘Let A and B be rings and U a(B,A)-bimodule.If BU is flat and UA is finitely generated projective(resp.,BU is finitely generated projective and UA is flat),then the characterizations of level modules and Gorenstein AC-projective modules(resp.,absolutely clean modules and Gorenstein AC-injective modules)over the formal triangular matrix ring T=(A0 UB)are given.As applications,it is proved that every Gorenstein AC-projective left T-module is projective if and only if each Gorenstein AC-projective left A-module and B-module is projective,and every Gorenstein AC-injective left T-module is injective if and only if each Gorenstein AC-injective left A-module and B-module is injective.Moreover,Gorenstein AC-projective and AC-injective dimensions over the formal triangular matrix ring T are studied.
基金supported by Fondazione Cariverona,Program“Ricerca Scientifica di Eccellenza 2018”(Project“Reducing Complexity in Algebra,Logic,Combinatorics-REDCOM”)supported by China Scholarship Council(Grant No.201906860022)。
文摘Ring epimorphisms often induce silting modules and cosilting modules,termed minimal silting or minimal cosilting.The aim of this paper is twofold.Firstly,we determine the minimal tilting and minimal cotilting modules over a tame hereditary algebra.In particular,we show that a large cotilting module is minimal if and only if it has an adic module as a direct summand.Secondly,we discuss the behavior of minimality under ring extensions.We show that minimal cosilting modules over a commutative noetherian ring extend to minimal cosilting modules along any flat ring epimorphism.Similar results are obtained for commutative rings of small homological dimensions.
基金This research is in part supported by a grant from IPM.
文摘Let R be a commutative Noetherian ring and p be a prime ideal of R such that the ideal pRp is principal and ht(p)≠0. In this note, the anthors describe the explicit structure of the injective envelope of the R-module R/p.
基金Specialized Research Fund (20050284015, 20030284033) for the Doctoral Program of Higher Education of China the Postdoctoral Research Fund (2005037713) of China Jiangsu Planned Projects for Postdoctoral Research Fund (0203003403) the Research Fund of Nanjing Institute of Technology of China
文摘In this paper, we introduce the concept of almost cotorsion modules. A module is called almost cotorsion if it is subisomorphic to its cotorsion envelope. Some characterizations of almost cotorsion modules are given. It is also proved that every module is a direct summand of an almost cotorsion module. As an application, perfect rings are characterized in terms of almost cotorsion modules.
