Cofinite Lifting And τ-Semiregular Modules

In homological algebra and modules theory, there are many dual concepts,suchas projectivity and injectivity, essential and superfluous submodules, injective hullsand projective covers. Two dual concept in formality are not bound to have cor-responding properties. Take injective hulls and projective covers for example, weall know that while every module has a injective hull it does not necessarily havea projective cover. In order to see which kind of modules have a projective cover,it is necessary to make a good study of lifting properties of modules. Therefore, inrecent years, as a generalization of projective supplemented modules, lifting moduleshave attracted comprehensive attentions. From the aspects of cofinite submodulesand finite generated submodules, lifting modules are generalized to cofinite liftingmodules andτ-semiregular modules respectively.In chapter 1 of this paper we first give the characterizations of cofinite liftingmodules. Then we discuss the problem about finite direct sums of cofinite liftingmodules and obtain a sufficient condition for that. At last, we investigate the de-composable properties of strong cofinite lifting modules, which is a generalization ofcofinite lifting modules.In chaper 2, firstly, we describle some equivalent definitions ofτ-semiregularmodules. Especially, when M is a projective module, the equivalent statements ofτ-semiregular modules are given. Secondly, we introduce the structure theoremof countably generatedτ-semiregular modules. Finally, we shift to study the finitedirect sums ofτ-semiregular modules.In chaper 3, we introduce the concepts of Serre -⊕-supplemented modulesand amply Serre-supplemented modules. By the means of Serre- subcategories,we do a research toward supplemented and lifting properties of submodules. Atfirst, we get that. the finite direct sums of Serre -⊕-supplemented modules are still Serre -⊕-supplemented modules, and so do the direct summand of Serre-⊕-supplemented modules under some coations. Then, we proof the images of amplySerre-supplemented modules are also amply Serre-supplemented modules. Atlast, we investigated the decomposable properties of strong Serre-lifting modules.