In basic homological algebra, the flat and injective dimensions of modules play an important and fundamental role. In this paper, the closely related IFP-flat and IFP-injective dimensions are introduced and studied. W...In basic homological algebra, the flat and injective dimensions of modules play an important and fundamental role. In this paper, the closely related IFP-flat and IFP-injective dimensions are introduced and studied. We show that IFP-fd(M) = IFP-id(M+) and IFP-fd(M+)=IFP-id(M) for any R-module M over any ring R. Let :Z-In (resp., "Zgv,~) he the class of all left (resp., right) R-modules of IFP-injective (resp., IFP-flat) dimension at most n. We prove that every right R-module has an IFn- preenvelope, (IFn,IF⊥n) is a perfect cotorsion theory over any ring R, and for any ring R with IFP-id(RR) 〈 n, (IIn,II⊥n) is a perfect cotorsion theory. This generalizes and improves the earlier work (J. Algebra 242 (2001), 447-459). Finally, some applications are given.展开更多
Let R be a ring. A fight R-module M is called f-projective if Ext^1 (M, N) = 0 for any f-injective right R-module N. We prove that (F-proj,F-inj) is a complete cotorsion theory, where (F-proj (F-inj) denotes th...Let R be a ring. A fight R-module M is called f-projective if Ext^1 (M, N) = 0 for any f-injective right R-module N. We prove that (F-proj,F-inj) is a complete cotorsion theory, where (F-proj (F-inj) denotes the class of all f-projective (f-injective) right R-modules. Semihereditary rings, von Neumann regular rings and coherent rings are characterized in terms of f-projective modules and f-injective modules.展开更多
Let R be a ring and S a class of R-modules. S-superfluous epimorphisms and S-essential monomorphisms are introduced and studied in this article. As applications, some new characterizations of von Neumann regular rings...Let R be a ring and S a class of R-modules. S-superfluous epimorphisms and S-essential monomorphisms are introduced and studied in this article. As applications, some new characterizations of von Neumann regular rings and perfect rings are given. Finally, these notions are also used to study minimal homomorphisms.展开更多
基金supported by National Natural Science Foundation of China(10961021,11001222)
文摘In basic homological algebra, the flat and injective dimensions of modules play an important and fundamental role. In this paper, the closely related IFP-flat and IFP-injective dimensions are introduced and studied. We show that IFP-fd(M) = IFP-id(M+) and IFP-fd(M+)=IFP-id(M) for any R-module M over any ring R. Let :Z-In (resp., "Zgv,~) he the class of all left (resp., right) R-modules of IFP-injective (resp., IFP-flat) dimension at most n. We prove that every right R-module has an IFn- preenvelope, (IFn,IF⊥n) is a perfect cotorsion theory over any ring R, and for any ring R with IFP-id(RR) 〈 n, (IIn,II⊥n) is a perfect cotorsion theory. This generalizes and improves the earlier work (J. Algebra 242 (2001), 447-459). Finally, some applications are given.
基金the Jiangsu Teachers University of Technology of China(No.Kyy06109)
文摘Let R be a ring. A fight R-module M is called f-projective if Ext^1 (M, N) = 0 for any f-injective right R-module N. We prove that (F-proj,F-inj) is a complete cotorsion theory, where (F-proj (F-inj) denotes the class of all f-projective (f-injective) right R-modules. Semihereditary rings, von Neumann regular rings and coherent rings are characterized in terms of f-projective modules and f-injective modules.
基金Supported by Specialized Research Fund for the Doctoral Program of Higher Education (20050284015)National Natural Science Foundation of China (10771096)
文摘Let R be a ring and S a class of R-modules. S-superfluous epimorphisms and S-essential monomorphisms are introduced and studied in this article. As applications, some new characterizations of von Neumann regular rings and perfect rings are given. Finally, these notions are also used to study minimal homomorphisms.
基金supported by the National Natural Science Foundation of China(10871042,10971024)the Specialized Research Fund for the Doctoral Program of Higher Education(200802860024)