Let C be a class of R-modules closed under isomorphisms and finite direct sums. It is first shown that the finite direct sum of almost C-precovers is an almost C-precover and the direct sum of an almost C-cover and a ...Let C be a class of R-modules closed under isomorphisms and finite direct sums. It is first shown that the finite direct sum of almost C-precovers is an almost C-precover and the direct sum of an almost C-cover and a weak C-cover is a weak C-cover. Then the notion of almost C-preenvelopes is introduced and studied.展开更多
We prove that, for any n 〉 2, the classes of FPn-injective modules and of FPn-flat modules are both covering and preenveloping over any ring R. This includes the case of FP∞-injective and FP∞-flat modules （i.e., a...We prove that, for any n 〉 2, the classes of FPn-injective modules and of FPn-flat modules are both covering and preenveloping over any ring R. This includes the case of FP∞-injective and FP∞-flat modules （i.e., absolutely clean and, respectively, level modules）. Then we consider a generalization of the class of （strongly） Gorenstein flat modules, i.e., the （strongly） Gorenstein AC-flat modules （cycles of exact complexes of flat modules that remain exact when tensored with any absolutely clean module）. We prove that some of the properties of Gorenstein fiat modules extend to the class of Gorenstein AC-flat modules; for example, we show that this class is precovering over any ring R. We also show that （as in the case of Gorenstein flat modules） every Gorenstein AC-flat module is a direct summand of a strongly Gorenstein AC-flat module. When R is such that the class of Gorenstein AC-flat modules is closed under extensions, the converse is also true. Moreover, we prove that if the class of Gorenstein AC-flat modules is closed under extensions, then it is covering.展开更多
An R-module M is called Gorenstein FP-injective if there is an exact sequence …→E1→E0→E^0→E^1→… of FP-injective R-modules with M=ker（E^0→E^1） and such that Hom（E,-） leaves the sequence exact whenever E is ...An R-module M is called Gorenstein FP-injective if there is an exact sequence …→E1→E0→E^0→E^1→… of FP-injective R-modules with M=ker（E^0→E^1） and such that Hom（E,-） leaves the sequence exact whenever E is an FP-injective R-module.Some properties of Gorenstein FP-injective are obtained.Moreover,it is proved that a ring is left Noetherian if and only if every Gorenstein FP-injective left R-module is Gorenstein injective.Furthermore,it is shown that over an n-FC and perfect ring R,a left R-module M is Gorenstein FP-injective if and only if MFH for some FP-injective left R-module F and some strongly Gorenstein FP-injective R-module H.In view of this,Gorenstein FP-injective precovers and Gorenstein FP-injective preenvelopes are considered.展开更多
基金This research was partially supported by the Specialized Research Fund for the Doctoral Program of Higher ducation of China (20020284009, 20030284033) NSF of China (10331030) Jiangsu Planned Projects for Postdoctoral Research Funds (0203003403)the Postdoctoral Research Funds of China (2006037713)and the Nanjing Institute of Technology of China.
文摘Let C be a class of R-modules closed under isomorphisms and finite direct sums. It is first shown that the finite direct sum of almost C-precovers is an almost C-precover and the direct sum of an almost C-cover and a weak C-cover is a weak C-cover. Then the notion of almost C-preenvelopes is introduced and studied.
文摘We prove that, for any n 〉 2, the classes of FPn-injective modules and of FPn-flat modules are both covering and preenveloping over any ring R. This includes the case of FP∞-injective and FP∞-flat modules （i.e., absolutely clean and, respectively, level modules）. Then we consider a generalization of the class of （strongly） Gorenstein flat modules, i.e., the （strongly） Gorenstein AC-flat modules （cycles of exact complexes of flat modules that remain exact when tensored with any absolutely clean module）. We prove that some of the properties of Gorenstein fiat modules extend to the class of Gorenstein AC-flat modules; for example, we show that this class is precovering over any ring R. We also show that （as in the case of Gorenstein flat modules） every Gorenstein AC-flat module is a direct summand of a strongly Gorenstein AC-flat module. When R is such that the class of Gorenstein AC-flat modules is closed under extensions, the converse is also true. Moreover, we prove that if the class of Gorenstein AC-flat modules is closed under extensions, then it is covering.
基金The National Natural Science Foundation of China (No.10971024)Specialized Research Fund for the Doctoral Program of Higher Education (No. 200802860024)
文摘An R-module M is called Gorenstein FP-injective if there is an exact sequence …→E1→E0→E^0→E^1→… of FP-injective R-modules with M=ker（E^0→E^1） and such that Hom（E,-） leaves the sequence exact whenever E is an FP-injective R-module.Some properties of Gorenstein FP-injective are obtained.Moreover,it is proved that a ring is left Noetherian if and only if every Gorenstein FP-injective left R-module is Gorenstein injective.Furthermore,it is shown that over an n-FC and perfect ring R,a left R-module M is Gorenstein FP-injective if and only if MFH for some FP-injective left R-module F and some strongly Gorenstein FP-injective R-module H.In view of this,Gorenstein FP-injective precovers and Gorenstein FP-injective preenvelopes are considered.
