In this paper, we mainly investigate some properties of strongly n-Gorenstein projective, injective and flat modules under the extension of rings, which mainly including excellent extensions, morita equivalences, poly...In this paper, we mainly investigate some properties of strongly n-Gorenstein projective, injective and flat modules under the extension of rings, which mainly including excellent extensions, morita equivalences, polynomial extensions and localizations.展开更多
This paper is a study of strongly Ding projective modules with respect to a semidualizing module. The class of strongly Ding flat modules with respect to a semidualizing module is also investigated, and the relationsh...This paper is a study of strongly Ding projective modules with respect to a semidualizing module. The class of strongly Ding flat modules with respect to a semidualizing module is also investigated, and the relationship between strongly Ding projective modules and strongly Ding flat modules with respect to a semidualizing module is characterized.Some well-known results on strongly Ding projective modules, n-strongly Ding projective modules and strongly D_C-projective modules are generalized and unified.展开更多
Let R be a graded ring. We define and study strongly Gorenstein gr-projective, gr-injective, and gr-flat modules. Some connections among these modules are discussed. We also explore the relations between the graded an...Let R be a graded ring. We define and study strongly Gorenstein gr-projective, gr-injective, and gr-flat modules. Some connections among these modules are discussed. We also explore the relations between the graded and the ungraded strongly Gorenstein modules.展开更多
There is a variety of nice results about strongly Gorenstein flat modules over coherent rings. These results are done by Ding, Lie and Mao. The aim of this paper is to generalize some of these results, and to give hom...There is a variety of nice results about strongly Gorenstein flat modules over coherent rings. These results are done by Ding, Lie and Mao. The aim of this paper is to generalize some of these results, and to give homological descriptions of the strongly Gorenstein flat dimension (of modules and rings) over arbitrary associative rings.展开更多
This article is concerned with the strongly Gorenstein flat dimensions of modules and rings.We show this dimension has nice properties when the ring is coherent,and extend the well-known Hilbert＇s syzygy theorem to t...This article is concerned with the strongly Gorenstein flat dimensions of modules and rings.We show this dimension has nice properties when the ring is coherent,and extend the well-known Hilbert＇s syzygy theorem to the strongly Gorenstein flat dimensions of rings.Also,we investigate the strongly Gorenstein flat dimensions of direct products of rings and（almost）excellent extensions of rings.展开更多
Strongly irreducible submodules of modules are defined as follows： A submodule N of an Rmodule M is said to be strongly irreducible if for submodules L and K of M, the inclusion L ∩ K ∈ N implies that either L ∈ N...Strongly irreducible submodules of modules are defined as follows： A submodule N of an Rmodule M is said to be strongly irreducible if for submodules L and K of M, the inclusion L ∩ K ∈ N implies that either L ∈ N or K ∈ N. The relationship among the families of irreducible, strongly irreducible, prime and primary submodules of an R-module M is considered, and a characterization of Noetherian modules which contain a non-prime strongly irreducible submodule is given.展开更多
The concepts of strongly lifting modules and strongly dual Rickart modules are introduced and their properties are studied and relations between them are given in this paper. It is shown that a strongly lifting module...The concepts of strongly lifting modules and strongly dual Rickart modules are introduced and their properties are studied and relations between them are given in this paper. It is shown that a strongly lifting module has the strongly summand sum property and the generalized Hopfian property, and a ring R is a strongly regular ring if and only if RR is a strongly dual Rickart module, if and only if aR is a fully invariant direct summand of RR for every a∈R.展开更多
Let R be a domain.In this paper,we show that if R is one dimensional,then R is a Noetherian Warfield domain if and only if every maximal ideal of R is 2-generated and for every maximal ideal M of R,M is divisorial in ...Let R be a domain.In this paper,we show that if R is one dimensional,then R is a Noetherian Warfield domain if and only if every maximal ideal of R is 2-generated and for every maximal ideal M of R,M is divisorial in the ring(M:M).We also prove that a Noetherian domain R is a Noetherian Warfield domain if and only if for every maximal ideal M of R,M^(2) can be generated by two elements.Finally,we give a sufficient condition under which all ideals of R are strongly Gorenstein projective.展开更多
In this paper, we study the closeness of strongly （∞）-hopfian properties under some constructions such as the ring of Morita context, direct products, triangular matrix, fraction ring etc. Also, we prove that if M[...In this paper, we study the closeness of strongly （∞）-hopfian properties under some constructions such as the ring of Morita context, direct products, triangular matrix, fraction ring etc. Also, we prove that if M[X] is strongly hopfian （resp. strongly co-hopfian） in R[X]-Mod, then M is strongly hopfian （resp. strongly co-hopfian） in R-Mod.展开更多
The introduction of w-operation in the class of flat modules has been successful. Let R be a ring. An R-module M is called a w-fiat module if Tor1r（M, N） is GV-torsion for all R-modules N. In this paper, we introduc...The introduction of w-operation in the class of flat modules has been successful. Let R be a ring. An R-module M is called a w-fiat module if Tor1r（M, N） is GV-torsion for all R-modules N. In this paper, we introduce the w-operation in Gorenstein homological algebra. An R-module M is called Ding w-flat if there exists an exact sequence of projective R-modules ... → P1 → P0 → p0 → p1 → ... such that M Im（P0 → p0） and such that the functor HomR （-,F） leaves the sequence exact whenever F is w-flat. Several well- known classes of rings are characterized in terms of Ding w-flat modules. Some examples are given to show that Ding w-flat modules lie strictly between projective modules and Gorenstein projective modules. The Ding w-flat dimension （of modules and rings） and the existence of Ding w-flat precovers are also studied.展开更多
基金Supported by the NNSF of China(10901129)Supported by the SRFDP(20096203120001)
文摘In this paper, we mainly investigate some properties of strongly n-Gorenstein projective, injective and flat modules under the extension of rings, which mainly including excellent extensions, morita equivalences, polynomial extensions and localizations.
基金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（KJ2017A040）
文摘This paper is a study of strongly Ding projective modules with respect to a semidualizing module. The class of strongly Ding flat modules with respect to a semidualizing module is also investigated, and the relationship between strongly Ding projective modules and strongly Ding flat modules with respect to a semidualizing module is characterized.Some well-known results on strongly Ding projective modules, n-strongly Ding projective modules and strongly D_C-projective modules are generalized and unified.
基金Acknowledgements This work was Foundation of China （Grant No. 11371187） Province of China （Grant No. BK20160771） supported by the National Natural Science and the Natural Science Foundation of Jiangsu
文摘Let R be a graded ring. We define and study strongly Gorenstein gr-projective, gr-injective, and gr-flat modules. Some connections among these modules are discussed. We also explore the relations between the graded and the ungraded strongly Gorenstein modules.
文摘There is a variety of nice results about strongly Gorenstein flat modules over coherent rings. These results are done by Ding, Lie and Mao. The aim of this paper is to generalize some of these results, and to give homological descriptions of the strongly Gorenstein flat dimension (of modules and rings) over arbitrary associative rings.
基金Supported by the National Natural Science Foundation of China (Grant No.10961021)
文摘This article is concerned with the strongly Gorenstein flat dimensions of modules and rings.We show this dimension has nice properties when the ring is coherent,and extend the well-known Hilbert＇s syzygy theorem to the strongly Gorenstein flat dimensions of rings.Also,we investigate the strongly Gorenstein flat dimensions of direct products of rings and（almost）excellent extensions of rings.
文摘Strongly irreducible submodules of modules are defined as follows： A submodule N of an Rmodule M is said to be strongly irreducible if for submodules L and K of M, the inclusion L ∩ K ∈ N implies that either L ∈ N or K ∈ N. The relationship among the families of irreducible, strongly irreducible, prime and primary submodules of an R-module M is considered, and a characterization of Noetherian modules which contain a non-prime strongly irreducible submodule is given.
基金Acknowledgements This work was supported by the Natural Science Foundation of Gansu Province （No. 1310RJZA029） and the Fundamental Research F~nds for the Universities in Gansu Province.
文摘The concepts of strongly lifting modules and strongly dual Rickart modules are introduced and their properties are studied and relations between them are given in this paper. It is shown that a strongly lifting module has the strongly summand sum property and the generalized Hopfian property, and a ring R is a strongly regular ring if and only if RR is a strongly dual Rickart module, if and only if aR is a fully invariant direct summand of RR for every a∈R.
基金This work was partially supported by the Department of Mathematics in Kyungpook National University and National Natural Science Foundation of China(Grant No.11671283)The second author was supported by the Basic Science Research Program through the National Research Foundation of Korea(NRF)funded by the Ministry of Education,Science and Technology(2017R1C1B1008085),Korea.
文摘Let R be a domain.In this paper,we show that if R is one dimensional,then R is a Noetherian Warfield domain if and only if every maximal ideal of R is 2-generated and for every maximal ideal M of R,M is divisorial in the ring(M:M).We also prove that a Noetherian domain R is a Noetherian Warfield domain if and only if for every maximal ideal M of R,M^(2) can be generated by two elements.Finally,we give a sufficient condition under which all ideals of R are strongly Gorenstein projective.
基金Supported by National Natural Science Foundation of China (Grant No. 10961021)
文摘In this paper, we study the closeness of strongly （∞）-hopfian properties under some constructions such as the ring of Morita context, direct products, triangular matrix, fraction ring etc. Also, we prove that if M[X] is strongly hopfian （resp. strongly co-hopfian） in R[X]-Mod, then M is strongly hopfian （resp. strongly co-hopfian） in R-Mod.
文摘The introduction of w-operation in the class of flat modules has been successful. Let R be a ring. An R-module M is called a w-fiat module if Tor1r（M, N） is GV-torsion for all R-modules N. In this paper, we introduce the w-operation in Gorenstein homological algebra. An R-module M is called Ding w-flat if there exists an exact sequence of projective R-modules ... → P1 → P0 → p0 → p1 → ... such that M Im（P0 → p0） and such that the functor HomR （-,F） leaves the sequence exact whenever F is w-flat. Several well- known classes of rings are characterized in terms of Ding w-flat modules. Some examples are given to show that Ding w-flat modules lie strictly between projective modules and Gorenstein projective modules. The Ding w-flat dimension （of modules and rings） and the existence of Ding w-flat precovers are also studied.
