Envelopes Covers And Cotorsion Theories

In 1981, Enochs abstractly gave the definitions of envelopes and covers of modules from injective envelopes and projective covers. In fact, they are left and right minimal approximations defined by Auslander in representation theory of algebras at the same time. The theory of envelopes and covers takes an important part in theory of rings and modules, homological algebra, representation theory of algebras and so on. Torsion theory was defined relative to Hom fuctor; cotorsion theory was defined relative to Ext functor. Cotorsion theory plays a crucial role in resolving Flat Cover Conjecture. It is a hot item of international research for the theory of envelopes and covers, cotorsion theory today. In this dissertation, depending on the recent results in the fields of homological dimensions, classic modules, envelopes and covers, we obtain new methods to characterize rings by applying cotorsion theory and so on.In Chapter 2, the F-dimension and C-dimension of modules in cotorsion theory (F, C) are introduced by Ext functor and deeply studied . The F- dimension unifies projrctive dimension, flat dimension and FP-projective dimension; the C-dimension unifies injection dimension and cotorsion dimension. On one hand, new characterizations of von Neumann regular rings, perfect rings are given. We also obtain new characterizations of flat dimension by using Ext functors. On the other hand, some recent results of Mao Lixin and Ding Nanqing published in Comm. Algrbra in 2005 are generalized.In Chapter 3, the existence of envelopes and covers is mainly discussed. We first study the relation between envelopes and covers in cotorsion theory. Some results of Enochs are obtained as corollaries. Furthermore, it is known that injective envelopes and projective covers defined by commutative diagrams are coincident with those defined by essential submodules and superfluous submodule, respectively. Hence F-essential submodules and F-superfluous submodules are introduced and studied. The relations between such modules and general envelopes (covers) are explored. The definition of F-essential submodules extends the definitions of essential submodules and pure essential submodules; the definition of F-superfluous submodules extends the definitions of superfluous submodules and weakly superfluous submodules. In the last part...