Ann-Projective Modules, Ann-Injective Modules And Ann-Flat Modules

Projective modules, injective modules and flat modules are main objects in homological algebra and module theory, which also play an important roll in algebraic geometry and algebraic K-theory. In this thesis, we mainly extend these three modules and discuss about them in three chapters to investigate the properties that they possess.In Chapter one, we introduce ann-exact sequences. After that ann-exact functors and ann-projective modules are defined by them. Through studying the properties of ann-projective modules, we obtain a series of useful theorems. Meanwhile, ann-projective resolution for an arbitrary left R-module M is defined, from which we get ann-projective dimensions and ann-projective global dimensions as well. Together with complex, we extend a very crucial theorem, Comparison Theorem, which is very important in the homological theory of rings, modules and complexes.In Chapter two, we introduce the notion of ann-injective modules and obtain many properties of it by comparing with injective modules. Also, it has been proved that every left R-module M has an ann-injective resolution, from which ann-injective dimensions and ann-injective global dimensions are defined.In the last chapter, the concept of flat modules is generalized to ann-flat modules. We list a series of equivalent characterizations of it and define ann-flat dimensions and ann-weak dimensions so as to give, a classification of related rings. Moreover, we disclose the relation between ann-flat modules and ann-injective modules, i.e., B is an ann-flat right R-module if and only if its characteristic module B~* = Hom_R(B, Q/Z) is ann-injective.