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 R be a commutative domain with 1 and Q(≠R)its field of quotients.In this note an R-module M is called w_(∞)-Warfield cotorsion if M∈WC∩P^(⊥)_(w_(∞)),where WC denotes the class of all Warfield cotorsion R-mod...Let R be a commutative domain with 1 and Q(≠R)its field of quotients.In this note an R-module M is called w_(∞)-Warfield cotorsion if M∈WC∩P^(⊥)_(w_(∞)),where WC denotes the class of all Warfield cotorsion R-modules and P_(w_(∞))the class of all w_(∞)-projective R-modules.It is shown that R is a PVMD if and only if all w-cotorsion R-modules are w_(∞)-Warfield cotorsion,and that R is a Krull domain if and only if every w-Matlis cotorsion strong w-module over R is a w_(∞)-Warfield cotorsion w-module.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 R be a ring and n,k be two non-negative integers.As an extension of several known notions,we introduce and study(n,k)-weak cotorsion modules using the class of right R-modules with n-weak fat dimensions at most k....Let R be a ring and n,k be two non-negative integers.As an extension of several known notions,we introduce and study(n,k)-weak cotorsion modules using the class of right R-modules with n-weak fat dimensions at most k.Various examples and applications are also given.展开更多
An integral domain R is called a locally almost perfect domain provided that Rm is an almost perfect domain for any maximal ideal m of R.In this paper,we give several characterizations of locally almost perfect domain...An integral domain R is called a locally almost perfect domain provided that Rm is an almost perfect domain for any maximal ideal m of R.In this paper,we give several characterizations of locally almost perfect domains in terms of locally perfect rings,almost projective modules,weak-injective modules,almost strongly flat modules and strongly Matlis cotorsion modules.展开更多
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This work was partially supported by the Sichuan Science and Technology Program(2023NSFSC0074)the National Natural Science Foundation of China(11961050,12061001)Aba Teachers University(ASS20230106,20210403005,20220301016).
文摘Let R be a commutative domain with 1 and Q(≠R)its field of quotients.In this note an R-module M is called w_(∞)-Warfield cotorsion if M∈WC∩P^(⊥)_(w_(∞)),where WC denotes the class of all Warfield cotorsion R-modules and P_(w_(∞))the class of all w_(∞)-projective R-modules.It is shown that R is a PVMD if and only if all w-cotorsion R-modules are w_(∞)-Warfield cotorsion,and that R is a Krull domain if and only if every w-Matlis cotorsion strong w-module over R is a w_(∞)-Warfield cotorsion w-module.
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.
文摘Let R be a ring and n,k be two non-negative integers.As an extension of several known notions,we introduce and study(n,k)-weak cotorsion modules using the class of right R-modules with n-weak fat dimensions at most k.Various examples and applications are also given.
基金Supported by the National Natural Science Foundation of China(Grant No.12061001).
文摘An integral domain R is called a locally almost perfect domain provided that Rm is an almost perfect domain for any maximal ideal m of R.In this paper,we give several characterizations of locally almost perfect domains in terms of locally perfect rings,almost projective modules,weak-injective modules,almost strongly flat modules and strongly Matlis cotorsion modules.
