Finite Cotorsion Modules And Characterizing Rings

Homological algebra was established in the middle of 1940s, and it is a important subject that is based on a series of research made by famous math-ematician S.Eilenberg and S.Maclane and so on. It's thoughts and methods mainly come from the homological theory of complex in algebraic topology. Rings and modules and the complex on them are the research object of ho-mological algebra. In the development process, homological algebra makes the best of the methods in category theory, using the functor Hom, (?) and their derived functor Ext, Tor as the basic means, giving effectively some homologi-cal invariants (homological dimension) of rings, so that the same class of rings has the same homological invariants, thus it provides a powerful new tool for the study of ring theory.An R-module M is called a cotorsion R-module, if Ext R 1(F, M)=0 for all flat R-module F. As the class of projective modules, injective modules and flat modules, the one of cotorsion modules is an important class of modules, and a major object of the homological algebra. There is a very close relation between the class of cotorsion modules with the other three ones especially the flat ones and injective ones. From the definition of cotorsion modules we see that cotorsion modules is accompanied by flat ones. Lots of properties of cotorsion modules are obtained with the help of flat modules. For instance, it is shown that all modules in any ring have flat covers and cotorsion envelopes, and any R- module has cotorsion envelopes if and only if it has flat covers. Furthermore, we find that cotorsion modules can be seen as the expansion of injective ones just as flat ones relative to projective ones. Studying the cotorsion modules can help us to understand the properties of various classes of modules and the relation between them. Particularly, in the course of studying the relation between the class of cotorsion modules and the others, we find that the cotorsion modules are very useful in characterizing several rings. However, our research of cotorsion modules is later than the others, which start in the 1980s, for no more than thirty years. So that the works about cotorsion modules are seldom, and the teaching materials for the graduate students of relevant major are scarce of this part. So the research of cotorsion modules is of very important sense in both practice and theory. This thesis round the title of cotorsion modules, make a overall summary about some relative concept and properties of cotorsion modules first, then extend the cotorsion modules and introduce a new concept:finite cotorsion modules. Let R be a ring, an R-module M is called a finite cotorsion module, if ExtR 1(F, M)= 0 for any finitely generated P-flat R-module F. Next, the author attempt to study the properties of the finite cotorsion modules. In the end, the author discuss the properties of finitely cotorsion on Noether rings. There are two nice properties on Noether rings:(a) For Noetherian rings, the submodule of every finitely generated modules is also finitely generated; (b) All ideals of a Noetherian ring are finitely generated. By restricting the ring R to a Noether one and by means of the preceding two properties, we get several new results and find that the finite cotorsion modules play important roles in characterizing some rings.