The Module-Theoretic Characterizations Of W-coherent Rings

Some module-theoretic characterizations of w-coherent rings are provided in this thesis.In Chapter 1,we recall some basic conceptions on w-theories,such as w-modules,w-envelopes,w-Noetherian rings,w-coherent rings,w-flat modules.In Chapter 2,the covering and enveloping properties for the classes of w-modules and w-flat modules are studied.Firstly,it is shown that the class of w-modules is covering and enveloping.Secondly,it is obtained that the class of w-flat modules is covering.Lastly,some new descriptions of VN-regular rings are given.In Chapter 3,the class w-F-ML of w-Mittag-Leffler modules over all flat modules is also introduced and studied.It is shown that R is w-Noetherian if and only if every R-module is in w-F-ML;R is w-coherent if and only if every(finitely generated)ideal of R is in w-F-ML.As an application,we give a new w-version of Chase Theorem.That is,R is w-coherent if and only if any direct product of flat R-modules is w-flat,if and only if any direct product of regular module R is w-flat.Examples of w-flat w-modules which are not flat are also given.In Chapter 4,absolutely purew-modules are utilized to characterize w-coherent rings.It is proved that R is w-coherent if and only if any directed limit of absolutely pure w-modules is an absolutely pure w-module,if and only if the class of absolutely pure w-modules is(pre)covering.In Chapter 5,the class w-?w-SS-? of-strictly ?w-stationary modules is introduced and studied.It is proved that R is w-Noetherian if and only if every R-module is in w-Sw-SS-?,if and only if every R-module is in w-?w-SS-?w;R is w-coherent if and only if every(finitely generated)ideal of R is in w-?w-SS-S,if and only if every(finitely generated)ideal of R is in w-?w-SS-?w.As an application,we give a w-version of the duality Theorem.That is,R is w-coherent if and only if HomR(M,E)is w-flat for any absolutely pure w-module M and any injective(w-)module E,if and only if HomR(M,E)is w-flat for any injective w-module M and any injective(w-)module E.