Let R be an associative ring with identity,and Y be a class of right R-modules,which contains all injective right R-modules.In this paper,we introduce the definition of Y-Gorenstein cotorsion modules,which is a genera...Let R be an associative ring with identity,and Y be a class of right R-modules,which contains all injective right R-modules.In this paper,we introduce the definition of Y-Gorenstein cotorsion modules,which is a generalization of cotorsion and Gorenstein cotorsion modules.We discuss the relationship between Gorenstein cotorsion,weakly Gorenstein cotorsion and Y-Gorenstein cotorsion modules.We investigate properties and characterizations of Y-Gorenstein cotorsion modules.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
Let A be an abelian category,and(X,Z,Y)be a complete hereditary cotorsion triple.We introduce the definition of n-Y-cotilting subcategories of A,and give a characterization of n-Y-cotilting subcategories,which is simi...Let A be an abelian category,and(X,Z,Y)be a complete hereditary cotorsion triple.We introduce the definition of n-Y-cotilting subcategories of A,and give a characterization of n-Y-cotilting subcategories,which is similar to Bazzoni characterization of n-cotilting modules.As an application,we prove that if GP is n-GI-cotilting over a virtually Gorenstein ring R,then R is an n-Gorenstein ring,where GP denotes the subcategory of Gorenstein projective R-modules and GI denotes the subcategory of Gorenstein injective R-modules.Furthermore,we investigate n-costar subcategories over arbitrary ring R,and the relationship between n-Icotilting subcategories with respect to cotorsion triple(P,R-Mod,I)and n-costar subcategories,where P denotes the subcategory of projective left R-modules and I denotes the subcategory of injective left R-modules.展开更多
In this paper, we introduce the definition of n-star subcategories, which is a generalization of n-star modules and n-C-star modules. We give some characterizations of n-star subcategories, and prove that M is an n-P-...In this paper, we introduce the definition of n-star subcategories, which is a generalization of n-star modules and n-C-star modules. We give some characterizations of n-star subcategories, and prove that M is an n-P-tilting subcategory with respect to cotorsion triple(P, R-Mod, I), if and only if M is an n-star subcategory with I ■Pres^(n)(M), where P denotes the subcategory of projective left R-modules and I denotes the subcategory of injective left R-modules.展开更多
Let R[P] be the one point extension of a k-algebra R by a projective R-module P.We prove that the extension of a complete ideal cotorsion pair in R-Mod is still a complete ideal cotorsion pair in R[P]-Mod.As an applic...Let R[P] be the one point extension of a k-algebra R by a projective R-module P.We prove that the extension of a complete ideal cotorsion pair in R-Mod is still a complete ideal cotorsion pair in R[P]-Mod.As an application,it is obtainable that the operation(-)_(m)[P]satisfies the so-called distributive law relating the operations of products and extensions of ideals under appropriate conditions.展开更多
The paper considers the problem of full transitivity of a cotorsion hull of a separable primary group G when a ring of endomorphisms E(G) of the group G has the form , where Es(G) is a subring of small endomorphisms o...The paper considers the problem of full transitivity of a cotorsion hull of a separable primary group G when a ring of endomorphisms E(G) of the group G has the form , where Es(G) is a subring of small endomorphisms of the ring E(G), whereas Jp is a ring of integer P-adic numbers. Investigation of the issue of full transitivity of a group is essentially helpful in studying its fully invariant subgroups as well as the lattice formed by these subgroups. It is proved that in the considered case, the cotorsion hull is not fully transitive. A lemma is proposed, which can be used in the study of full transitivity of a group and in other cases.展开更多
The paper considers the lattice of fully invariant subgroups of the cotorsion hull ?when a separable primary group T?is an arbitrary direct sum of torsion-complete groups.The investigation of this problem in the case ...The paper considers the lattice of fully invariant subgroups of the cotorsion hull ?when a separable primary group T?is an arbitrary direct sum of torsion-complete groups.The investigation of this problem in the case of a cotorsion hull is important because endomorphisms in this class of groups are completely defined by their action on the torsion part and for mixed groups the ring of endomorphisms is isomorphic to the ring of endomorphisms of the torsion part if and only if the group is a fully invariant subgroup of the cotorsion hull of its torsion part. In the considered case, the cotorsion hull is not fully transitive and hence it is necessary to introduce a new function which differs from an indicator and assigns an infinite matrix to each element of the cotorsion hull. The relation ?difined on the set ?of these matrices is different from the relation proposed by the autor in the countable case and better discribes the lower semilattice. The use of the relation ?essentially simplifies the verification of the required properties. It is proved that the lattice of fully invariant subgroups of the group is isomorphic to the lattice of filters of the lower semilattice.展开更多
Let H be a finite-dimensional weak Hopf algebra over a field κ and A an associative algebra, and A#;H a weak crossed product. In this paper, a spectral sequence for Ext is constructed which yields an estimate for cot...Let H be a finite-dimensional weak Hopf algebra over a field κ and A an associative algebra, and A#;H a weak crossed product. In this paper, a spectral sequence for Ext is constructed which yields an estimate for cotorsion dimension of A#;H in terms of the corresponding data for H and A.展开更多
Let C be a triangulated category. We define m-term subcategories on C induced by n-rigid subcategories, which are extriangulated subcategories of C. Then we give a one-to-one correspondence between cotorsion pairs on ...Let C be a triangulated category. We define m-term subcategories on C induced by n-rigid subcategories, which are extriangulated subcategories of C. Then we give a one-to-one correspondence between cotorsion pairs on 2-term subcategories G and support τ-tilting subcategories on an abelian quotient of G. If an m-term subcategory is induced by a co-t-structure, then we have a one-to-one correspondence between cotorsion pairs on it and cotorsion pairs on C under certain conditions.展开更多
Let H be a finite dimensional weak Hopf algebra and A be an Hmodule algebra.In this paper,we mainly discuss the relations of cotorsion dimension and FP-projective dimension between A#H and A.As applications,sufficient...Let H be a finite dimensional weak Hopf algebra and A be an Hmodule algebra.In this paper,we mainly discuss the relations of cotorsion dimension and FP-projective dimension between A#H and A.As applications,sufficient conditions are given for LCD(A#H)=LCD(A)and LFPD(A#H)=LFPD(A).展开更多
This paper focuses on a question raised by Holm and Jorgensen,who asked if the induced cotorsion pairs(Φ(X),Φ(X)^(⊥))and(^(⊥)Ψ(Y),Ψ(Y))in Rep(Q,A)—the category of all A-valued representations of a quiver Q—are...This paper focuses on a question raised by Holm and Jorgensen,who asked if the induced cotorsion pairs(Φ(X),Φ(X)^(⊥))and(^(⊥)Ψ(Y),Ψ(Y))in Rep(Q,A)—the category of all A-valued representations of a quiver Q—are complete whenever(X,Y)is a complete cotorsion pair in an abelian category A satisfying some mild conditions.We give an affirmative answer if the quiver Q is rooted.As an application,we show under certain mild conditions that if a subcategory L,which is not necessarily closed under direct summands,of A is special precovering(resp.,preenveloping),thenΦ(L)(resp.,Ψ(L))is special precovering(resp.,preenveloping)in Rep(Q,A).展开更多
Let R be a ring and let be the class of strongly Gorenstein fiat right R-modules. We call a right R-module M a weak Gorenstein cotorsion module if M is in the class ⊥. Properties of weak Gorenstein cotorsion modu...Let R be a ring and let be the class of strongly Gorenstein fiat right R-modules. We call a right R-module M a weak Gorenstein cotorsion module if M is in the class ⊥. Properties of weak Gorenstein cotorsion modules are investigated. It is shown that weak Gorenstein cotorsion R-modules over coherent rings are indeed weaker than Gorenstein cotorsion R-modules. Weak Gorenstein cotorsion dimension for modules and rings are also studied.展开更多
Let n≥1 be an integer.In this article,we study the notions of Cartan-Eilenberg FP_(n)injective and Cartan-Eilenberg FP_(n)-flat complexes.We also investigate duality pairs relative to the Cartan-Eilenberg FP_(n)-inje...Let n≥1 be an integer.In this article,we study the notions of Cartan-Eilenberg FP_(n)injective and Cartan-Eilenberg FP_(n)-flat complexes.We also investigate duality pairs relative to the Cartan-Eilenberg FP_(n)-injective and Cartan-Eilenberg FP_(n)-flat complexes.As applications,we obtain that any complex admits Cartan-Eilenberg FP_(n)-injective(resp.,Cartan-Eilenberg FP_(n)-flat)cover and preenvelope and we characterize left n-coherent rings using these two classes of complexes.展开更多
基金Supported by the 2020 Scienti c Research Projects in Universities of Gansu Province(Grant No.2020A-277).
文摘Let R be an associative ring with identity,and Y be a class of right R-modules,which contains all injective right R-modules.In this paper,we introduce the definition of Y-Gorenstein cotorsion modules,which is a generalization of cotorsion and Gorenstein cotorsion modules.We discuss the relationship between Gorenstein cotorsion,weakly Gorenstein cotorsion and Y-Gorenstein cotorsion modules.We investigate properties and characterizations of Y-Gorenstein cotorsion modules.
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.
基金Supported by Research Project in Institutions of Higher Learning in Gansu Province(Grant No.2019B-224)Innovation Fund Project of Colleges and Universities in Gansu Province(Grant No.2020A-277)。
文摘Let A be an abelian category,and(X,Z,Y)be a complete hereditary cotorsion triple.We introduce the definition of n-Y-cotilting subcategories of A,and give a characterization of n-Y-cotilting subcategories,which is similar to Bazzoni characterization of n-cotilting modules.As an application,we prove that if GP is n-GI-cotilting over a virtually Gorenstein ring R,then R is an n-Gorenstein ring,where GP denotes the subcategory of Gorenstein projective R-modules and GI denotes the subcategory of Gorenstein injective R-modules.Furthermore,we investigate n-costar subcategories over arbitrary ring R,and the relationship between n-Icotilting subcategories with respect to cotorsion triple(P,R-Mod,I)and n-costar subcategories,where P denotes the subcategory of projective left R-modules and I denotes the subcategory of injective left R-modules.
基金Supported by the 2020 Scientific Research Projects in Universities of Gansu Province (Grant No. 2020A-277)。
文摘In this paper, we introduce the definition of n-star subcategories, which is a generalization of n-star modules and n-C-star modules. We give some characterizations of n-star subcategories, and prove that M is an n-P-tilting subcategory with respect to cotorsion triple(P, R-Mod, I), if and only if M is an n-star subcategory with I ■Pres^(n)(M), where P denotes the subcategory of projective left R-modules and I denotes the subcategory of injective left R-modules.
基金Supported by Zhejiang Provincial Natural Science Foundation of China(LY18A010032)
文摘Let R[P] be the one point extension of a k-algebra R by a projective R-module P.We prove that the extension of a complete ideal cotorsion pair in R-Mod is still a complete ideal cotorsion pair in R[P]-Mod.As an application,it is obtainable that the operation(-)_(m)[P]satisfies the so-called distributive law relating the operations of products and extensions of ideals under appropriate conditions.
文摘The paper considers the problem of full transitivity of a cotorsion hull of a separable primary group G when a ring of endomorphisms E(G) of the group G has the form , where Es(G) is a subring of small endomorphisms of the ring E(G), whereas Jp is a ring of integer P-adic numbers. Investigation of the issue of full transitivity of a group is essentially helpful in studying its fully invariant subgroups as well as the lattice formed by these subgroups. It is proved that in the considered case, the cotorsion hull is not fully transitive. A lemma is proposed, which can be used in the study of full transitivity of a group and in other cases.
文摘The paper considers the lattice of fully invariant subgroups of the cotorsion hull ?when a separable primary group T?is an arbitrary direct sum of torsion-complete groups.The investigation of this problem in the case of a cotorsion hull is important because endomorphisms in this class of groups are completely defined by their action on the torsion part and for mixed groups the ring of endomorphisms is isomorphic to the ring of endomorphisms of the torsion part if and only if the group is a fully invariant subgroup of the cotorsion hull of its torsion part. In the considered case, the cotorsion hull is not fully transitive and hence it is necessary to introduce a new function which differs from an indicator and assigns an infinite matrix to each element of the cotorsion hull. The relation ?difined on the set ?of these matrices is different from the relation proposed by the autor in the countable case and better discribes the lower semilattice. The use of the relation ?essentially simplifies the verification of the required properties. It is proved that the lattice of fully invariant subgroups of the group is isomorphic to the lattice of filters of the lower semilattice.
基金The NSF(KJ2016A545,KJ2015B12,2017ZR08zd)of Anhui Provincethe key projectsoutstanding young talent support program(gxyq ZD2016353)of Anhui Province
文摘Let H be a finite-dimensional weak Hopf algebra over a field κ and A an associative algebra, and A#;H a weak crossed product. In this paper, a spectral sequence for Ext is constructed which yields an estimate for cotorsion dimension of A#;H in terms of the corresponding data for H and A.
基金supported by the National Natural Science Foundation of China(Grant No.12171397)Panyue Zhou is supported by the National Natural Science Foundation of China(Grant No.12371034)by the Hunan Provincial Natural Science Foundation of China(Grant No.2023JJ30008)。
文摘Let C be a triangulated category. We define m-term subcategories on C induced by n-rigid subcategories, which are extriangulated subcategories of C. Then we give a one-to-one correspondence between cotorsion pairs on 2-term subcategories G and support τ-tilting subcategories on an abelian quotient of G. If an m-term subcategory is induced by a co-t-structure, then we have a one-to-one correspondence between cotorsion pairs on it and cotorsion pairs on C under certain conditions.
文摘Let H be a finite dimensional weak Hopf algebra and A be an Hmodule algebra.In this paper,we mainly discuss the relations of cotorsion dimension and FP-projective dimension between A#H and A.As applications,sufficient conditions are given for LCD(A#H)=LCD(A)and LFPD(A#H)=LFPD(A).
基金partly supported by NSF of China(Grant No.11971388)partly supported by NSF of China(Grant No.12171146)+4 种基金partly supported by NSF of China(Grant No.12271230)partly supported by NSF of China(Grant No.12171297)the Scientific Research Funds of Huaqiao University(Grant No.605-50Y22050)the Fujian Alliance Of Mathematics(Grant No.2024SXLMMS04)the Foundation for Innovative Fundamental Research Group Project of Gansu Province(Grant No.23JRRA684)。
文摘This paper focuses on a question raised by Holm and Jorgensen,who asked if the induced cotorsion pairs(Φ(X),Φ(X)^(⊥))and(^(⊥)Ψ(Y),Ψ(Y))in Rep(Q,A)—the category of all A-valued representations of a quiver Q—are complete whenever(X,Y)is a complete cotorsion pair in an abelian category A satisfying some mild conditions.We give an affirmative answer if the quiver Q is rooted.As an application,we show under certain mild conditions that if a subcategory L,which is not necessarily closed under direct summands,of A is special precovering(resp.,preenveloping),thenΦ(L)(resp.,Ψ(L))is special precovering(resp.,preenveloping)in Rep(Q,A).
基金The authors wish to express their sincere thanks to the referees for their valuable comments and suggestions. The first author was supported by the Postdoctoral Science Foundation of China （2017M611851）, the Jiangsu Planned Projects for Postdoctoral Research Funds （1601151C） and the Provincial Natural Science Foundation of Anhui Province of China （KJ2017A040）. The second author was supported by the NSFC （11771212）, and the first two authors were supported by a project funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions. The third author was supported by the NSFC （11501257, 11671069, 11771212） and the Postdoctoral Science Foundation of China （2016M600426）.
文摘Let R be a ring and let be the class of strongly Gorenstein fiat right R-modules. We call a right R-module M a weak Gorenstein cotorsion module if M is in the class ⊥. Properties of weak Gorenstein cotorsion modules are investigated. It is shown that weak Gorenstein cotorsion R-modules over coherent rings are indeed weaker than Gorenstein cotorsion R-modules. Weak Gorenstein cotorsion dimension for modules and rings are also studied.
基金Supported by the National Natural Science Foundation of China(Grant No.12061061)the Science and Technology Project of Gansu Province(Grant No.22JR5RA153).
文摘Let n≥1 be an integer.In this article,we study the notions of Cartan-Eilenberg FP_(n)injective and Cartan-Eilenberg FP_(n)-flat complexes.We also investigate duality pairs relative to the Cartan-Eilenberg FP_(n)-injective and Cartan-Eilenberg FP_(n)-flat complexes.As applications,we obtain that any complex admits Cartan-Eilenberg FP_(n)-injective(resp.,Cartan-Eilenberg FP_(n)-flat)cover and preenvelope and we characterize left n-coherent rings using these two classes of complexes.
